didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Compare commits

base: didericis/math-research:b605931678fea832ef401d3af82dab5bdfd9d3df
Branches Tags
didericis/math-research:master didericis/math-research:split-medial-pigeonhole-programme
..
compare: didericis/math-research:split-medial-pigeonhole-programme
Branches Tags
didericis/math-research:master didericis/math-research:split-medial-pigeonhole-programme

1 Commits
b605931678 .. split-medial-pigeonhole-programme

Author SHA1 Message Date
didericis d3fc4bfc4c Split medial pigeonhole programme into its own paper 
Move Section 5 of "Medial Tire Decompositions of Plane Triangulations"
into a new standalone paper, "The Medial Pigeonhole Programme", which
cites the medial tire paper for its terminology and notation. Convert
the three cross-references that pointed into earlier sections (annular
teeth, bite-face-count, boundary medial vertices) into citations.

Remove Section 5 from the medial tire paper and update its abstract to
drop the moved chain-pigeonhole claim, pointing to the follow-up paper.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 21:08:06 -04:00
29 changed files with 1210 additions and 3646 deletions
Expand all files Collapse all files
papers/medial_pigeonhole_programme/paper.aux
+28
View File
@@ -0,0 +1,28 @@
\relax
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A medial pigeonhole programme}}{1}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{2.1}{2}}
\newlabel{conj:medial-chain-pigeonhole}{{2.2}{2}}
\newlabel{conj:medial-route-fct}{{2.3}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Kempe-cycle conservation across medial tires}}{2}{}\protected@file@percent }
\newlabel{lem:kempe-cycles}{{3.1}{2}}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\newlabel{lem:kempe-conservation}{{3.2}{3}}
\newlabel{def:kempe-balanced}{{3.3}{3}}
\newlabel{rem:kempe-balance-necessary}{{3.4}{3}}
\bibcite{bauerfeld-medial-tire}{1}
\bibcite{bauerfeld-nested-tire-decompositions}{2}
\bibcite{tait-original}{3}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent }
\gdef \@abspage@last{6}
papers/medial_pigeonhole_programme/paper.fdb_latexmk
+135
View File
@@ -0,0 +1,135 @@
# Fdb version 3
["pdflatex"] 1781485473 "paper.tex" "paper.pdf" "paper" 1781485473
"/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm" 1136768653 1332 1fde11373e221473104d6cc5993f046e ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss10.tfm" 1136768653 1316 b636689f1933f24d1294acdf6041daaa ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss8.tfm" 1136768653 1296 d77f431d10d47c8ea2cc18cf45346274 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb" 1248133631 32166 b0c356b15f19587482a9217ce1d8fa67 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb" 1248133631 24457 5cbb7bdf209d5d1ce9892a9b80a307cc ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 ""
"/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1601326656 992 855ff26741653ab54814101ca36e153c ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1601326656 43820 1fef971b75380574ab35a0d37fd92608 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1601326656 19324 f4e4c6403dd0f1605fd20ed22fa79dea ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1601326656 6038 ccb406740cc3f03bbfb58ad504fe8c27 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1601326656 6944 e12f8f7a7364ddf66f93ba30fb3a3742 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1601326656 4883 42daaf41e27c3735286e23e48d2d7af9 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1601326656 2544 8c06d2a7f0f469616ac9e13db6d2f842 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1601326656 44195 5e390c414de027626ca5e2df888fa68d ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1601326656 17311 2ef6b2e29e2fc6a2fc8d6d652176e257 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex" 1601326656 21302 788a79944eb22192a4929e46963a3067 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex" 1601326656 9690 01feb7cde25d4293ef36eef45123eb80 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1601326656 33335 dd1fa4814d4e51f18be97d88bf0da60c ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1601326656 2965 4c2b1f4e0826925746439038172e5d6f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex" 1601326656 5196 2cc249e0ee7e03da5f5f6589257b1e5b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex" 1601326656 20726 d4c8db1e2e53b72721d29916314a22ea ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex" 1601326656 35249 abd4adf948f960299a4b3d27c5dddf46 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex" 1601326656 21989 fdc867d05d228316de137a9fc5ec3bbe ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1601326656 8893 e851de2175338fdf7c17f3e091d94618 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex" 1601326656 4572 4a19637ef65ce88ad2f2d5064b69541d ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex" 1608933718 11518 738408f795261b70ce8dd47459171309 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1621110968 186007 6e7dfe0bd57520fd5f91641aa72dcac8 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1601326656 32995 ac577023e12c0e4bd8aa420b2e852d1a ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex" 1557692582 3063 8c415c68a0f3394e45cfeca0b65f6ee6 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex" 1601326656 521 8e224a7af69b7fee4451d1bf76b46654 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1601326656 13391 84d29568c13bdce4133ab4a214711112 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1601326656 104935 184ed87524e76d4957860df4ce0cd1c3 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1601326656 10165 cec5fa73d49da442e56efc2d605ef154 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex" 1601326656 28178 41c17713108e0795aac6fef3d275fbca ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1601326656 9989 c55967bf45126ff9b061fa2ca0c4694f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex" 1601326656 3865 ac538ab80c5cf82b345016e474786549 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1557692582 3177 27d85c44fbfe09ff3b2cf2879e3ea434 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1621110968 11024 0179538121bc2dba172013a3ef89519f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1608933718 7854 4176998eeefd8745ac6d2d4bd9c98451 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1601326656 3379 781797a101f647bab82741a99944a229 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1601326656 92405 f515f31275db273f97b9d8f52e1b0736 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex" 1601326656 37376 11cd75aac3da1c1b152b2848f30adc14 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex" 1601326656 8471 c2883569d03f69e8e1cabfef4999cfd7 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1601326656 21201 08d231a2386e2b61d64641c50dc15abd ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex" 1601326656 16121 346f9013d34804439f7436ff6786cef7 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex" 1621110968 44784 cedaa399d15f95e68e22906e2cc09ef8 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex" 1621110968 465 d68603f8b820ea4a08cce534944db581 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg" 1601326656 926 2963ea0dcf6cc6c0a770b69ec46a477b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def" 1601326656 5546 f3f24d7898386cb7daac70bdd2c4d6dc ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def" 1601326656 12601 4786e597516eddd82097506db7cfa098 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex" 1621110968 61163 9b2eefc24e021323e0fc140e9826d016 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex" 1601326656 1896 b8e0ca0ac371d74c0ca05583f6313c91 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex" 1601326656 7778 53c8b5623d80238f6a20aa1df1868e63 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex" 1606168878 23997 a4bed72405fa644418bea7eac2887006 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex" 1621110968 37060 797782f0eb50075c9bc952374d9a659a ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex" 1601326656 37431 9abe862035de1b29c7a677f3205e3d9f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex" 1601326656 4494 af17fb7efeafe423710479858e42fa7e ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex" 1601326656 7251 fb18c67117e09c64de82267e12cd8aa4 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex" 1621110968 29274 e15c5b7157d21523bd9c9f1dfa146b8e ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def" 1621110968 6825 a2b0ea5b539dda0625e99dd15785ab59 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty" 1636758526 4128 8eea906621b6639f7ba476a472036bbe ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty" 1636758526 2444 926f379cc60fcf0c6e3fee2223b4370d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def" 1601931164 19103 48d29b6e2a64cb717117ef65f107b404 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty" 1622581934 18399 7e40f80366dffb22c0e7b70517db5cb4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty" 1636758526 7996 a8fb260d598dcaf305a7ae7b9c3e3229 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty" 1622581934 2671 4de6781a30211fe0ea4c672e4a2a8166 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty" 1636758526 4009 187ea2dc3194cd5a76cd99a8d7a6c4d0 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty" 1601326656 1090 bae35ef70b3168089ef166db3e66f5b2 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty" 1601326656 410 615550c46f918fcbee37641b02a862d9 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty" 1601326656 21013 f4ff83d25bb56552493b030f27c075ae ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty" 1601326656 989 c49c8ae06d96f8b15869da7428047b1e ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty" 1601326656 339 c2e180022e3afdb99c7d0ea5ce469b7d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty" 1601326656 306 c56a323ca5bf9242f54474ced10fca71 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty" 1601326656 443 8c872229db56122037e86bcda49e14f3 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty" 1601326656 348 ee405e64380c11319f0e249fed57e6c5 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty" 1601326656 274 5ae372b7df79135d240456a1c6f2cf9a ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty" 1601326656 325 f9f16d12354225b7dd52a3321f085955 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty" 1635798903 56029 3f7889dab51d620aa43177c391b7b190 ""
"/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 ""
"/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 ""
"/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc ""
"/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
"paper.aux" 1781485473 1312 7ba44acc160ff1167e676415e34169c0 "pdflatex"
"paper.tex" 1781485463 17511 c613c597a654f07644b998ea69f242e5 ""
(generated)
"paper.aux"
"paper.log"
"paper.pdf"
papers/medial_pigeonhole_programme/paper.fls
+471
View File
@@ -0,0 +1,471 @@
PWD /Users/didericis/Code/math-research/papers/medial_pigeonhole_programme
INPUT /usr/local/texlive/2022/texmf.cnf
INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf
INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt
INPUT paper.tex
OUTPUT paper.log
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT ./paper.aux
INPUT paper.aux
INPUT paper.aux
OUTPUT paper.aux
INPUT /usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmss8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
OUTPUT paper.pdf
INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm
INPUT paper.aux
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb
papers/medial_tire_cuts/paper.log → papers/medial_pigeonhole_programme/paper.log
+27 -31
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 15 JUN 2026 01:07
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 JUN 2026 21:05
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -495,39 +495,35 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
e
))
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
LaTeX Warning: `h' float specifier changed to `ht'.
[2] [3] (./whole_medial_seed59_min5.tikz) [4] [5] (./paper.aux) )
[2] [3] [4] [5] [6] (./paper.aux) )
Here is how much of TeX's memory you used:
13803 strings out of 478268
275582 string characters out of 5846347
655790 words of memory out of 5000000
31627 multiletter control sequences out of 15000+600000
477661 words of font info for 60 fonts, out of 8000000 for 9000
13536 strings out of 478268
270232 string characters out of 5846347
537626 words of memory out of 5000000
31366 multiletter control sequences out of 15000+600000
476880 words of font info for 57 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191
84i,15n,89p,907b,804s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/te
xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/tex
live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx6.pfb></usr/local/texli
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx7.pfb></usr/local/texlive
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/20
22/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022
/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/te
xmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf
-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-di
st/fonts/type1/public/amsfonts/cm/cmss10.pfb></usr/local/texlive/2022/texmf-dis
t/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist
/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/f
onts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fo
nts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/font
s/type1/public/amsfonts/cm/cmtt10.pfb>
Output written on paper.pdf (5 pages, 226751 bytes).
84i,7n,89p,414b,277s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/ty
pe1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
e1/public/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
blic/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
ic/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
nts/cm/cmss10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
s/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
m/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/sym
bols/msam10.pfb>
Output written on paper.pdf (6 pages, 211901 bytes).
PDF statistics:
103 PDF objects out of 1000 (max. 8388607)
63 compressed objects within 1 object stream
101 PDF objects out of 1000 (max. 8388607)
62 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
13 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/medial_pigeonhole_programme/paper.pdf
BIN
View File
Binary file not shown.
papers/medial_pigeonhole_programme/paper.tex
+442
View File
@@ -0,0 +1,442 @@
%% filename: amsart-template.tex
%% American Mathematical Society
%% AMS-LaTeX v.2 template for use with amsart
%% ====================================================================
\documentclass{amsart}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\begin{document}
\title{The Medial Pigeonhole Programme}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Kempe chain, Four Colour Theorem}
\date{}
\dedicatory{}
\begin{abstract}
Building on the medial tire decomposition of a plane triangulation, we
formulate a pigeonhole programme for the Four Colour Theorem in medial
terms. Each tire carries a boundary-state restriction relation, and a
proper vertex $3$-colouring of the full medial graph is a compatible
selection of these boundary states across the tire tree. We state a
chain-pigeonhole conjecture asserting that the restriction relations
cannot remain mutually disjoint along every branch, and we refine the
boundary states by recording how two-colour Kempe cycles are routed
through each annular tire region. This yields a Kempe-enhanced
restriction relation and a notion of Kempe-compatible gluing along level
cycles.
\end{abstract}
\maketitle
\section{Introduction}
This paper continues the medial tire programme begun
in~\cite{bauerfeld-medial-tire}. We use freely the terminology and
notation introduced there. For a plane triangulation $G$ with fixed
embedding, $M(G)$ denotes the full medial graph, and the tire-tree
decomposition $\mathcal{T}(G,S)$ at a level source $S$
of~\cite{bauerfeld-nested-tire-decompositions} induces a decomposition
of $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one for each
tread $T$, glued along their boundary medial vertex sets
$\partial_{\mathrm{out}}\mathsf{M}(T)$ and
$\partial_{\mathrm{in}}\mathsf{M}(T)$. We also use the annular medial
cycle $A(T)$, its up and down teeth and their apexes, the bites and the
auxiliary plane graph $B(T)$, and the medial tire restriction relation
$R_T$ of~\cite{bauerfeld-medial-tire}.
By the Tait--medial correspondence of~\cite{bauerfeld-medial-tire},
proper vertex $3$-colourings of $M(G)$ are in natural bijection with
proper $3$-edge-colourings of the cubic planar dual $G^*$. Thus the
Four Colour Theorem is the assertion that the full medial graph of every
plane triangulation is properly vertex $3$-colourable, and the medial
tire decomposition turns this into a question about how local boundary
colourings compose across the tire tree.
\section{A medial pigeonhole programme}
The restriction relation $R_T$ records exactly the local information
needed to pass a medial $3$-colouring through a tire. In a nested
chain
\[
T_0 \supset T_1 \supset \cdots \supset T_k,
\]
the outer boundary state of $T_{i+1}$ must match an inner boundary
state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.
\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
subgraph induced by that boundary set, considered up to permutation of
the three colours and the dihedral symmetries of the boundary walk
when that boundary is a cycle.
\end{definition}
\begin{conjecture}[Medial chain-pigeonhole principle]
\label{conj:medial-chain-pigeonhole}
There is a function $N(k)$ such that the following holds. Let
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
tire treads whose relevant boundary medial walks have length at most
$k$. Then two adjacent restriction relations in the chain have
compatible medial boundary states after colour permutation and boundary
symmetry. Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint proper vertex $3$-colouring
restrictions.
\end{conjecture}
\begin{conjecture}[Medial tire route to the Four Colour Theorem]
\label{conj:medial-route-fct}
For every plane triangulation $G$ and every level source $S$, the
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
compatible selection of boundary states across the tire tree. Hence
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
\end{conjecture}
\begin{remark}
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
the Four Colour Theorem when combined with Tait's correspondence. The
point of the formulation is not to weaken the target theorem, but to
move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}
\section{Kempe-cycle conservation across medial tires}
We now record an additional structure carried by proper
$3$-colourings of medial graphs. This structure will be useful for
describing how colourings glue across level cycles.
Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
\varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$. For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.
Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
cycle. We call these components the $P$-Kempe cycles of $\varphi$.
\begin{lemma}[Kempe chains are cycles]
\label{lem:kempe-cycles}
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$. For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}
\begin{proof}
Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.
In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$. Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$. It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$. Hence each
component of $M_P$ is a cycle.
\end{proof}
Let $T$ be a medial tire region. We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.
The following lemma is the basic conservation principle.
\begin{lemma}[Kempe-cycle conservation across level cycles]
\label{lem:kempe-conservation}
Let $C$ be a level cycle of $M$ separating a parent side from a child
side. Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
of $C$ without also leaving it.
Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}
\begin{proof}
By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side. Consider the intersection of $K$ with one of these
regions. Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$. Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
We now use these Kempe cycles to single out the colourings of a full
medial tire graph that respect the annular tooth structure.
\begin{definition}[Kempe-balanced colouring]
\label{def:kempe-balanced}
Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
the subgraph induced by the vertices of colours $a$ and $b$. Since
$\mathsf{M}(T)$ need not be $4$-regular, the components of
$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly
one $P$-Kempe chain.
A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
face of $B(T)$ that is not a tooth---namely the root face or a bite
inner-gap face in the sense of~\cite{bauerfeld-medial-tire}. The
\emph{tooth apexes incident to} a valid face $F$ are:
\begin{itemize}
\item the up-tooth apexes (\cite{bauerfeld-medial-tire}), when
$F$ is the outer face;
\item the singleton down-tooth apexes whose annular edge lies on $F$,
when $F$ is interior---the apex on annular edge $m$ being incident to
the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
is none.
\end{itemize}
Bite apexes are never incident to a valid face in this sense.
For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
lie on a $P$-Kempe chain. The colouring $\varphi$ is
\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
every colour pair $P$.
\end{definition}
\begin{remark}[Necessity of Kempe-balance]
\label{rem:kempe-balance-necessary}
A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
$3$-colouring of the whole medial graph $M(G)$ only when it is
Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
condition at some valid face and colour pair cannot extend to a proper
$3$-colouring of $M(G)$. This is an instance of Kempe-cycle
conservation (Lemma~\ref{lem:kempe-conservation}). The tooth apexes
incident to a valid face are boundary medial vertices
(\cite{bauerfeld-medial-tire}) lying on a single level
cycle of the tire decomposition: the up-tooth apexes lie on the outer
level cycle, and the singleton down-tooth apexes incident to an interior
non-tooth face lie on the inner level cycle bounding that face. In the
$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
meets each level cycle in an even number of $P$-coloured incidences; for
a given valid face these incidences are exactly its incident tooth
apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
This argument is verified computationally. For bite-free pieces---capped
triangulated annuli on annular cycles of length $6,8,10,12$---every proper
$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring. The same
holds for pieces carrying a bite, including the case where singleton down
teeth lie in the bite's inner-gap face: there the inner level cycle splits
into a child level cycle per gap, and conservation across each child cycle
supplies the parity (in the checked example the three singleton down apexes
of a bite gap are a rainbow in every restriction).
\end{remark}
More generally, let $T$ be a medial tire region with boundary
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.
\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$. Let
\[
\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
\item a boundary colouring
\[
\alpha:V(\mathcal C(T))\to\{1,2,3\};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
a pairing $\pi_P$ of the $P$-coloured boundary incidences of
$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}
Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
\kappa_T(\varphi).
\]
\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
\mathcal K_T
=
\left\{
\kappa_T(\varphi):
\varphi \text{ is a proper }3\text{-colouring of } M(T)
\right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}
The annular structure of a tire is useful in two distinct ways. First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices. Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire. Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs. The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.
\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.
Let
\[
\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
\in \mathcal K_T
\]
and
\[
\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
\in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
\item the boundary colourings agree on $C$:
\[
\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
$P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}
In these terms, gluing local colourings is not merely a matter of
matching boundary colours. The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle. The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.
For a tire with one outer level cycle and several inner level cycles,
\[
\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles. The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously. Thus one should view a medial tire as a
multi-output transition operator
\[
\mathcal K_T:
C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.
The guiding principle is therefore:
\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}
\begin{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-nested-tire-decompositions}
E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},
Proceedings of the Royal Society of Edinburgh \textbf{10} (1880),
729--729.
\end{thebibliography}
\end{document}
papers/medial_tire_cuts/experiments/deepcap_seed7_cap.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 50 KiB

papers/medial_tire_cuts/experiments/deepcap_seed7_dual.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 227 KiB

papers/medial_tire_cuts/experiments/deepcap_seed7_tires.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 297 KiB

papers/medial_tire_cuts/experiments/draw_medial_tire_cut.py
-286
View File
@@ -1,286 +0,0 @@
"""Draw the walk-depth labelling and cut of a medial tire decomposition.
Paper-graphics companion to ``run_medial_tire_cut_experiment.py``: it imports
``run_experiment`` from there, runs the pipeline on a random maximal planar
graph, and emits TikZ. By default it draws one ``tikzpicture`` (walk-depth
labels + cut slits) per recognised full medial tire graph, using ``to_tikz``
from ``medial_tire_cut_labelling``. With ``--whole`` it instead draws a
two-panel Figure 3 graphic: the source graph with its source highlighted and
the whole medial graph M(G) drawn with every medial vertex at the midpoint of
its source edge and labelled by that source edge, with the full BFS-level chain
shown and the currently computed walk-depth labels and cuts marked.
This script only renders; the experiment itself draws nothing. Run with the
repo venv (networkx): ``.venv/bin/python``.
Examples:
.venv/bin/python draw_medial_tire_cut.py -n 20 --seed 59 > panels.tex
.venv/bin/python draw_medial_tire_cut.py -n 20 --seed 59 --whole > whole.tex
"""
from __future__ import annotations
import argparse
import math
import os
import sys
import networkx as nx
import numpy as np
_HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, _HERE)
from run_medial_tire_cut_experiment import run_experiment # noqa: E402
from medial_tire_cut_labelling import to_tikz # noqa: E402
from tire_realization_analysis import triangular_faces # noqa: E402
def tikz_panels(n: int, seed: int, scale: float = 1.6,
min_degree: int = 5, attempts: int = 1000) -> tuple[dict, list[str]]:
"""Run the experiment and return ``(result, panels)``, one TikZ panel per
recognised tread, each showing that tread's walk-depth labelling and cut."""
result = run_experiment(n=n, seed=seed, min_degree=min_degree, attempts=attempts)
panels = []
for d in sorted(result["results"]):
rec = result["results"][d]
panels.append(to_tikz(rec["g"], depth=rec["depth"], cuts=rec["cuts"],
entry_edge=rec["entry_edge"], scale=scale))
return result, panels
# --------------------------------------------------------------------------- #
# Figure 3: the source graph and midpoint drawing of the whole medial graph.
# --------------------------------------------------------------------------- #
def _source_layout(G: nx.Graph) -> dict[int, tuple[float, float]]:
"""Straight-line planar layout for the source graph, normalised to the unit
box and reused by the medial drawing."""
faces, _ = triangular_faces(G)
outer = list(faces[0])
outer_set = set(outer)
raw = {}
for i, v in enumerate(outer):
angle = math.radians(90.0 - i * 360.0 / len(outer))
raw[v] = np.array([math.cos(angle), math.sin(angle)], dtype=float)
inner = [v for v in sorted(G.nodes()) if v not in outer_set]
if inner:
idx = {v: i for i, v in enumerate(inner)}
n = len(inner)
A = np.zeros((n, n))
bx = np.zeros(n)
by = np.zeros(n)
for i, v in enumerate(inner):
nbrs = list(G.neighbors(v))
A[i, i] = 1.0
for w in nbrs:
if w in idx:
A[i, idx[w]] -= 1.0 / len(nbrs)
else:
bx[i] += raw[w][0] / len(nbrs)
by[i] += raw[w][1] / len(nbrs)
xs = np.linalg.solve(A, bx)
ys = np.linalg.solve(A, by)
for v in inner:
raw[v] = np.array([xs[idx[v]], ys[idx[v]]], dtype=float)
pts = np.array([raw[v] for v in G.nodes()], dtype=float)
center = 0.5 * (pts.max(axis=0) + pts.min(axis=0))
span = float(max(*(pts.max(axis=0) - pts.min(axis=0)), 1.0))
return {
v: tuple((raw[v] - center) / span)
for v in G.nodes()
}
def _edge_midpoint(pos: dict, edge) -> tuple[float, float]:
u, v = edge
ux, uy = pos[u]
vx, vy = pos[v]
return (0.5 * (ux + vx), 0.5 * (uy + vy))
def _edge_label(edge) -> str:
u, v = edge
return f"${u}\\!{{-}}\\!{v}$"
def _source_graph_tikz(result: dict, pos: dict, scale: float) -> str:
G, source = result["G"], result["source"]
L = []
A = L.append
A(f"\\begin{{tikzpicture}}[scale={scale},")
A(" sedge/.style={black!50, line width=0.35pt},")
A(" sv/.style={circle, draw=black!60, fill=white, inner sep=1.1pt},")
A(" srcv/.style={circle, draw=blue!75!black, fill=blue!18, line width=0.7pt, inner sep=1.8pt}]")
def pt(v):
x, y = pos[v]
return f"({x:.3f},{y:.3f})"
for u, v in sorted(G.edges()):
A(f"\\draw[sedge] {pt(u)}--{pt(v)};")
for v in sorted(G.nodes()):
style = "srcv" if v == source else "sv"
A(f"\\node[{style}] at {pt(v)} {{}};")
sx, sy = pos[source]
A(f"\\node[font=\\scriptsize, text=blue!70!black] at ({sx:.3f},{sy - 0.085:.3f}) {{source {source}}};")
A("\\end{tikzpicture}")
return "\n".join(L)
def _medial_midpoint_tikz(result: dict, pos: dict, scale: float) -> str:
"""Draw M(G) with each medial vertex at the midpoint of its source edge.
Every medial vertex is labelled by its source edge; same-level source edges
show the BFS level-chain tooth layers, and interlevel source edges show the
annular layers. Currently computed tire walk-depth labels and cut labels
are overlaid without moving the medial vertices away from their source
edges."""
G, M = result["G"], result["M"]
levels = nx.single_source_shortest_path_length(G, result["source"])
medial_pos = {edge: _edge_midpoint(pos, edge) for edge in M.nodes()}
apex_roles = {}
apex_walks = {}
for r in result["labels"]:
apex_roles[r["apex"]] = r["role"]
apex_walks.setdefault(r["apex"], []).append(r["walk"])
cut_records = []
cut_number = 1
for c in result.get("cap_cuts", []):
cut_records.append((cut_number, c["medial_vertex"], "cap", c))
cut_number += 1
for d in sorted(result["results"]):
rec = result["results"][d]
g, bij = rec["g"], rec["bij"]
for c in rec["cuts"]:
if c.vertex is None:
continue
cut_records.append((cut_number, bij[f"a{c.vertex}"], d, c))
cut_number += 1
L = []
A = L.append
A(f"\\begin{{tikzpicture}}[scale={scale},")
A(" base/.style={black!12, line width=0.25pt},")
A(" med/.style={black!38, line width=0.32pt},")
A(" annv/.style={circle, draw=black!70, fill=black!18, inner sep=1.0pt},")
A(" levone/.style={circle, draw=orange!75!black, fill=orange!20, inner sep=1.2pt},")
A(" levtwo/.style={circle, draw=violet!70!black, fill=violet!18, inner sep=1.2pt},")
A(" levthree/.style={circle, draw=teal!70!black, fill=teal!18, inner sep=1.2pt},")
A(" knownv/.style={circle, draw=red!70!black, fill=red!24, inner sep=1.5pt},")
A(" elbl/.style={font=\\tiny, text=black!70, inner sep=0.2pt},")
A(" dlbl/.style={font=\\tiny\\bfseries, text=black, inner sep=0.5pt},")
A(" cut/.style={red!80!black, line width=1.0pt},")
A(" cutlbl/.style={font=\\tiny, text=red!75!black}]")
def pt_med(edge):
x, y = medial_pos[edge]
return f"({x:.3f},{y:.3f})"
def pt_src(v):
x, y = pos[v]
return f"({x:.3f},{y:.3f})"
for u, v in sorted(result["G"].edges()):
A(f"\\draw[base] {pt_src(u)}--{pt_src(v)};")
for u, v in M.edges():
A(f"\\draw[med] {pt_med(u)}--{pt_med(v)};")
def chain_style(edge):
u, v = edge
lu, lv = levels[u], levels[v]
if lu != lv:
return "annv"
if edge in apex_roles:
return "knownv"
return {1: "levone", 2: "levtwo", 3: "levthree"}.get(lu, "annv")
for mv in sorted(M.nodes()):
A(f"\\node[{chain_style(mv)}] at {pt_med(mv)} {{}};")
for mv in sorted(M.nodes()):
x, y = medial_pos[mv]
A(f"\\node[elbl] at ({x:.3f},{y:.3f}) [yshift=-4.8pt] {{{_edge_label(mv)}}};")
for mv in sorted(apex_walks):
x, y = medial_pos[mv]
label = ",".join(str(w) for w in sorted(apex_walks[mv]))
A(f"\\node[dlbl] at ({x:.3f},{y:.3f}) [yshift=5.0pt] {{{label}}};")
for number, mv, _d, _cut in cut_records:
u, v = mv
ux, uy = pos[u]
vx, vy = pos[v]
mx, my = medial_pos[mv]
ex, ey = vx - ux, vy - uy
length = math.hypot(ex, ey) or 1.0
dx, dy = -0.035 * ey / length, 0.035 * ex / length
A(f"\\draw[cut] ({mx - dx:.3f},{my - dy:.3f})--({mx + dx:.3f},{my + dy:.3f});")
A(f"\\node[cutlbl] at ({mx + 2.4 * dx:.3f},{my + 2.4 * dy:.3f}) {{cut {number}}};")
A("\\end{tikzpicture}")
return "\n".join(L)
def medial_tikz(result: dict, scale: float = 7.0) -> str:
"""Two-panel TikZ for Figure 3: the source graph and the midpoint drawing of
its medial graph with all medial vertices labelled, plus the tire
walk-depth labels and cuts."""
pos = _source_layout(result["G"])
source = _source_graph_tikz(result, pos, scale=0.58 * scale)
medial = _medial_midpoint_tikz(result, pos, scale=scale)
return "\n".join([
"\\begin{tabular}{c}",
source,
"\\\\[-0.25ex]",
"{\\scriptsize source graph $G$}",
"\\\\[1.0ex]",
medial,
"\\\\[-0.25ex]",
"{\\scriptsize medial graph $M(G)$ at edge midpoints}",
"\\end{tabular}",
])
def main() -> None:
parser = argparse.ArgumentParser(description=__doc__,
formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("-n", type=int, default=20)
parser.add_argument("--seed", type=int, default=72)
parser.add_argument("--scale", type=float, default=1.6)
parser.add_argument("--min-degree", type=int, default=5,
help="reject random graphs below this minimum degree")
parser.add_argument("--attempts", type=int, default=1000,
help="number of consecutive seeds to try for --min-degree")
parser.add_argument("--whole", action="store_true",
help="draw the whole medial graph M(G) with all cuts, "
"instead of one panel per tread")
args = parser.parse_args()
if args.whole:
result = run_experiment(n=args.n, seed=args.seed,
min_degree=args.min_degree, attempts=args.attempts)
treads = sorted(result["results"])
print(f"% whole medial graph: n={args.n} seed={args.seed} "
f"graph_seed={result['graph_seed']} min_degree={result['min_degree']} "
f"source={result['source']} recognised treads={treads} "
f"|M(G)|={result['M'].number_of_nodes()}")
print(medial_tikz(result, scale=args.scale if args.scale != 1.6 else 7.0))
return
result, panels = tikz_panels(args.n, args.seed, scale=args.scale,
min_degree=args.min_degree, attempts=args.attempts)
treads = sorted(result["results"])
print(f"% medial tire cut: n={args.n} seed={args.seed} "
f"graph_seed={result['graph_seed']} min_degree={result['min_degree']} "
f"source={result['source']} recognised treads={treads}")
if not panels:
print("% (no recognised full medial tire graphs for this graph)")
for d, panel in zip(treads, panels):
g = result["results"][d]["g"]
print(f"% --- tread {d}: |A(T)|={g.n} word={g.tooth_word} "
f"bites={sorted(g.bites)} ---")
print(panel)
if __name__ == "__main__":
main()
papers/medial_tire_cuts/experiments/funcD_seed7_tread0.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 63 KiB

papers/medial_tire_cuts/experiments/medial_tire_cut_labelling.py
-388
View File
@@ -1,388 +0,0 @@
"""Walk-depth labelling and cut of a full medial tire graph.
Implements the procedure of Definition 2.1 ("Walk-depth labelling and cut") of
the *Medial Tire Cuts* paper:
1. Pick an arbitrary up tooth, the entry tooth; it has walk depth d.
2. Traverse all teeth bounding the inner face incident to the entry tooth
clockwise until reaching the entry tooth, incrementing the walk depth by 1
for each tooth traversed.
3. On reaching the last tooth in the face, perform a cut by duplicating the
annular vertex at which the traversal closes (the annular vertex shared by
the last tooth and the closing tooth).
4. Find the tooth t of highest walk depth that is a member of a bite.
5. If t is incident to a face F with unlabelled teeth, traverse the teeth of F
starting from t in the direction of the unlabelled tooth incident to t
(sharing an annular vertex), incrementing the walk depth as you go.
6. Repeat steps 3-5 until all teeth are labelled.
The full medial tire graph model (annular cycle A(T), up/down teeth, bites, the
auxiliary plane graph B(T) and its inner faces) is the one from the companion
``full_medial_tire_generator.py`` of the medial tire decompositions paper, which
we import.
Teeth are identified with the annular edges that carry them: edge i sits on the
annular vertices a_i and a_{(i+1) mod n} and carries exactly one tooth. A bite
(i, j) carries two teeth, one on edge i and one on edge j, that share the bite
apex p. The inner non-tooth faces of B(T) are the root face (written ``None``)
and one inner-gap face per bite.
"""
from __future__ import annotations
import argparse
import math
import os
import sys
# Import the full medial tire model from the companion paper's experiments.
_GEN_DIR = os.path.normpath(os.path.join(
os.path.dirname(__file__), "..", "..",
"medial_tire_decompositions_of_plane_triangulations", "experiments",
))
sys.path.insert(0, _GEN_DIR)
from full_medial_tire_generator import ( # noqa: E402
FullMedialTireGraph,
has_incident_bite,
innermost_bite,
satisfies_bite_face_condition,
)
Face = "tuple[int, int] | None" # a bite (i, j), or None for the root face
# ---------------------------------------------------------------------------
# Face structure of B(T).
# ---------------------------------------------------------------------------
def parent_face(graph: FullMedialTireGraph, bite: tuple[int, int]) -> Face:
"""The face directly enclosing ``bite``: the minimal-span bite strictly
containing it, or the root face ``None``."""
i, j = bite
enclosing = [b for b in graph.bites if b[0] < i and b[1] > j]
if not enclosing:
return None
return min(enclosing, key=lambda b: b[1] - b[0])
def door_bite(graph: FullMedialTireGraph, edge: int) -> tuple[int, int] | None:
"""The bite that ``edge`` is a door of (i.e. a bite edge), or None."""
for b in graph.bites:
if edge in b:
return b
return None
def faces_bordered(graph: FullMedialTireGraph, edge: int) -> list[Face]:
"""The inner non-tooth faces whose boundary the tooth on ``edge`` lies on.
A bite door borders two faces (its bite's gap and that bite's parent); any
other tooth borders the single face directly containing its edge.
"""
bite = door_bite(graph, edge)
if bite is not None:
return [bite, parent_face(graph, bite)]
return [innermost_bite(edge, graph.bites)]
def face_boundary(graph: FullMedialTireGraph, face: Face) -> list[int]:
"""The teeth (annular edges) bounding ``face``, in clockwise cyclic order.
Clockwise is increasing edge index. For the root face the boundary is read
around the whole cycle; for a bite gap (i, j) it is read along the arc
i, i+1, ..., j and closes through the bite apex. Edges enclosed by a child
bite are skipped (they belong to the child's gap face).
"""
n = graph.n
arc = range(n) if face is None else range(face[0], face[1] + 1)
return [k for k in arc if face in faces_bordered(graph, k)]
def all_faces(graph: FullMedialTireGraph) -> list[Face]:
return [None] + sorted(graph.bites)
def shared_annular_vertex(graph: FullMedialTireGraph, e1: int, e2: int) -> int | None:
"""The annular vertex a_k shared by edges ``e1`` and ``e2``, or None."""
n = graph.n
common = {e1, (e1 + 1) % n} & {e2, (e2 + 1) % n}
return next(iter(common)) if common else None
# ---------------------------------------------------------------------------
# The walk-depth labelling and cut.
# ---------------------------------------------------------------------------
class Cut:
"""A cut performed when a face traversal closes: the duplicated annular
vertex, together with the last labelled tooth and the closing tooth that
share it, and the face being closed."""
__slots__ = ("vertex", "last_tooth", "closing_tooth", "face", "order")
def __init__(self, vertex, last_tooth, closing_tooth, face, order):
self.vertex = vertex
self.last_tooth = last_tooth
self.closing_tooth = closing_tooth
self.face = face
self.order = order
def __repr__(self):
f = "root" if self.face is None else f"bite{self.face}"
return (f"Cut(order={self.order}, a{self.vertex}, "
f"last=e{self.last_tooth}, closing=e{self.closing_tooth}, face={f})")
def label_and_cut(graph: FullMedialTireGraph, entry_edge: int,
start_depth: int = 0) -> tuple[dict[int, int], list[Cut]]:
"""Run the procedure starting from up tooth ``entry_edge``.
Returns ``(depth, cuts)`` where ``depth`` maps each annular edge (tooth) to
its walk depth, and ``cuts`` is the list of cuts in the order performed.
"""
if graph.tooth_word[entry_edge] != "U":
raise ValueError(f"entry edge {entry_edge} is not an up tooth")
depth: dict[int, int] = {}
cuts: list[Cut] = []
counter = start_depth
def traverse(face: Face, start_edge: int, is_entry: bool) -> None:
nonlocal counter
boundary = face_boundary(graph, face)
m = len(boundary)
pos = boundary.index(start_edge)
if is_entry:
depth[start_edge] = counter
counter += 1
direction = +1
else:
# head toward the unlabelled tooth incident to the door t
direction = +1 if boundary[(pos + 1) % m] not in depth else -1
last_new = start_edge
i = pos
while True:
i = (i + direction) % m
edge = boundary[i]
if edge in depth: # the closing tooth
cuts.append(Cut(
vertex=shared_annular_vertex(graph, last_new, edge),
last_tooth=last_new, closing_tooth=edge,
face=face, order=len(cuts),
))
return
depth[edge] = counter
counter += 1
last_new = edge
# Steps 1-3: the entry face.
traverse(innermost_bite(entry_edge, graph.bites), entry_edge, is_entry=True)
# Steps 4-6: descend (or ascend) through bites, deepest first. The root
# face is ``None``, so we use a distinct sentinel for "no unlabelled face".
_MISSING = object()
while len(depth) < graph.n:
labelled_bite_teeth = sorted(
(e for e in depth if door_bite(graph, e) is not None),
key=lambda e: depth[e], reverse=True,
)
for t in labelled_bite_teeth:
target = next((F for F in faces_bordered(graph, t)
if any(e not in depth for e in face_boundary(graph, F))),
_MISSING)
if target is not _MISSING:
traverse(target, t, is_entry=False)
break
else:
break # no progress possible
return depth, cuts
# ---------------------------------------------------------------------------
# TikZ rendering.
# ---------------------------------------------------------------------------
def _coords(graph: FullMedialTireGraph,
r_ann=1.0, r_up=1.46, r_down=0.60) -> dict[str, tuple[float, float]]:
n = graph.n
def ang(k): # a_0 at the top, increasing k clockwise
return math.radians(90.0 - k * 360.0 / n)
def edge_mid_dir(i): # angle of the bisector of edge i's two endpoints
a0, a1 = ang(i), ang((i + 1) % n)
return math.atan2(math.sin(a0) + math.sin(a1), math.cos(a0) + math.cos(a1))
pos = {f"a{k}": (r_ann * math.cos(ang(k)), r_ann * math.sin(ang(k)))
for k in range(n)}
for i in graph.up_edges:
a = edge_mid_dir(i)
pos[f"u{i}"] = (r_up * math.cos(a), r_up * math.sin(a))
for i in graph.singleton_down_edges:
a = edge_mid_dir(i)
pos[f"d{i}"] = (r_down * math.cos(a), r_down * math.sin(a))
for (i, j) in graph.bites:
pts = [pos[f"a{i}"], pos[f"a{(i + 1) % n}"],
pos[f"a{j}"], pos[f"a{(j + 1) % n}"]]
cx = sum(p[0] for p in pts) / 4.0
cy = sum(p[1] for p in pts) / 4.0
pos[f"p{i}_{j}"] = (0.9 * cx, 0.9 * cy)
return pos
def _edge_midpoint(pos, graph, edge):
n = graph.n
a, b = pos[f"a{edge}"], pos[f"a{(edge + 1) % n}"]
return (0.5 * (a[0] + b[0]), 0.5 * (a[1] + b[1]))
def to_tikz(graph: FullMedialTireGraph,
depth: dict[int, int] | None = None,
cuts: list[Cut] | None = None,
entry_edge: int | None = None,
scale: float = 2.2) -> str:
"""A standalone ``tikzpicture`` for ``graph``; if ``depth`` is given, draw
the walk-depth labels and (with ``cuts``) the cut marks."""
pos = _coords(graph)
n = graph.n
L = []
A = L.append
A(f"\\begin{{tikzpicture}}[scale={scale},")
A(" ann/.style={circle, fill=black, inner sep=1.0pt},")
A(" upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},")
A(" downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},")
A(" bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},")
A(" cyc/.style={black, line width=1.0pt},")
A(" tth/.style={black!55, line width=0.4pt},")
A(" lbl/.style={font=\\scriptsize},")
A(" dlbl/.style={font=\\scriptsize\\bfseries, text=black},")
A(" cut/.style={red!80!black, line width=1.3pt},")
A(" cutlbl/.style={font=\\tiny, text=red!75!black}]")
def pt(name):
x, y = pos[name]
return f"({x:.3f},{y:.3f})"
# annular cycle
cyc = "--".join(pt(f"a{k}") for k in range(n)) + "--cycle"
A(f"\\draw[cyc] {cyc};")
# spokes
for i in graph.up_edges:
A(f"\\draw[tth] {pt(f'u{i}')}--{pt(f'a{i}')} {pt(f'u{i}')}--{pt(f'a{(i+1)%n}')};")
for i in graph.singleton_down_edges:
A(f"\\draw[tth] {pt(f'd{i}')}--{pt(f'a{i}')} {pt(f'd{i}')}--{pt(f'a{(i+1)%n}')};")
for (i, j) in graph.bites:
apex = f"p{i}_{j}"
for e in (i, j):
A(f"\\draw[tth] {pt(apex)}--{pt(f'a{e}')} {pt(apex)}--{pt(f'a{(e+1)%n}')};")
# vertices
for k in range(n):
A(f"\\node[ann] at {pt(f'a{k}')} {{}};")
for i in graph.up_edges:
A(f"\\node[upv] at {pt(f'u{i}')} {{}};")
for i in graph.singleton_down_edges:
A(f"\\node[downv] at {pt(f'd{i}')} {{}};")
for (i, j) in sorted(graph.bites):
A(f"\\node[bitev] at {pt(f'p{i}_{j}')} {{}};")
# walk-depth labels: placed along the spoke from apex toward the edge mid
if depth is not None:
for edge in range(n):
apex = graph.apex_of_edge(edge)
ax, ay = pos[apex]
mx, my = _edge_midpoint(pos, graph, edge)
f = 0.5
lx, ly = ax + f * (mx - ax), ay + f * (my - ay)
A(f"\\node[dlbl] at ({lx:.3f},{ly:.3f}) {{{depth[edge]}}};")
# cut marks: a short red slit across the duplicated annular vertex
if cuts:
for c in cuts:
if c.vertex is None:
continue
vx, vy = pos[f"a{c.vertex}"]
rad = math.atan2(vy, vx)
dx, dy = 0.16 * math.cos(rad), 0.16 * math.sin(rad)
A(f"\\draw[cut] ({vx-dx:.3f},{vy-dy:.3f})--({vx+dx:.3f},{vy+dy:.3f});")
lx, ly = vx + 0.30 * math.cos(rad), vy + 0.30 * math.sin(rad)
A(f"\\node[cutlbl] at ({lx:.3f},{ly:.3f}) {{cut {c.order+1}}};")
if entry_edge is not None:
ex, ey = pos[graph.apex_of_edge(entry_edge)]
rad = math.atan2(ey, ex)
tx, ty = ex + 0.34 * math.cos(rad), ey + 0.34 * math.sin(rad)
A(f"\\node[lbl, text=blue!60!black] at ({tx:.3f},{ty:.3f}) {{entry}};")
A("\\end{tikzpicture}")
return "\n".join(L)
# ---------------------------------------------------------------------------
# Worked example and CLI.
# ---------------------------------------------------------------------------
def worked_example() -> FullMedialTireGraph:
"""A clean 8-tooth piece: one bite (0,4), three down singletons 1,2,3 in its
gap, three up teeth 5,6,7 in the root face."""
return FullMedialTireGraph(n=8, tooth_word="DDDDDUUU", bites=frozenset({(0, 4)}))
def _check(graph: FullMedialTireGraph) -> None:
assert not has_incident_bite(graph.bites, graph.n), "bite uses incident edges"
assert satisfies_bite_face_condition(graph.tooth_word, graph.bites), \
"violates the bite-face condition"
assert graph.tooth_word.count("U") >= 3, "fewer than three up teeth"
def _describe(graph, depth, cuts) -> str:
lines = ["edge type walk-depth"]
for e in range(graph.n):
t = graph.tooth_word[e]
kind = {"U": "up"}.get(t, "down")
if door_bite(graph, e) is not None:
kind = "bite"
lines.append(f" e{e} {kind:<5} {depth[e]}")
lines.append("cuts (in order):")
for c in cuts:
f = "root" if c.face is None else f"bite{c.face}"
lines.append(f" cut {c.order+1}: duplicate a{c.vertex} "
f"(closing tooth e{c.closing_tooth} of {f})")
return "\n".join(lines)
def main() -> None:
parser = argparse.ArgumentParser(description=__doc__,
formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("--entry", default="u5",
help="entry up tooth, as an edge index or apex name like u5")
parser.add_argument("--start-depth", type=int, default=0)
parser.add_argument("--tikz", choices=["plain", "labelled", "both"],
help="emit TikZ for the worked example")
args = parser.parse_args()
entry = args.entry
edge = int(entry[1:]) if isinstance(entry, str) and entry.startswith("u") else int(entry)
graph = worked_example()
_check(graph)
depth, cuts = label_and_cut(graph, edge, start_depth=args.start_depth)
if args.tikz == "plain":
print(to_tikz(graph))
elif args.tikz == "labelled":
print(to_tikz(graph, depth=depth, cuts=cuts, entry_edge=edge))
elif args.tikz == "both":
print("% --- plain ---")
print(to_tikz(graph))
print("% --- labelled + cut ---")
print(to_tikz(graph, depth=depth, cuts=cuts, entry_edge=edge))
else:
print(f"worked example: n={graph.n} word={graph.tooth_word} "
f"bites={sorted(graph.bites)} entry=e{edge}")
print(_describe(graph, depth, cuts))
if __name__ == "__main__":
main()
papers/medial_tire_cuts/experiments/medial_tire_dual_cut_experiment.py
-765
View File
@@ -1,765 +0,0 @@
"""Source-dual cut from a chained medial tire cut.
Companion to ``run_medial_tire_cut_experiment.py``. Where that script reports
the cut graph of M(G), this one takes the same chained walk-depth labelling and
cut and reads it off as a *source-dual* cut: the planar dual of the source
triangulation G with the cut edges removed, as drawn for
``seed59_min5_dual_cut_1.png``.
The dual of a plane triangulation G has one node per triangular face and one
edge per primal edge (joining the two faces that share it). Its faces are the
*vertices* of G, each bounded by ``deg(v)`` dual edges. A medial tire cut at an
annular medial vertex removes the dual edge of the corresponding primal edge;
the interesting quantity is how many of those removed (``missing``) dual edges
surround each dual face (vertex of G). For ``seed59`` at source 5 the maximum
is 3, around the degree-9 vertex 3.
The level source is chosen by deep embedding: pick a random face of G, take the
deep embedding G' relative to that face (subdividing every neutral face,
including the chosen one), and use the outer-cap vertex x* placed inside the
chosen face as the source. The whole dual cut is then read off G'.
Four chained entry points (broad to narrow control):
* ``random_dual_cut(n, ...)`` -- find a random maximal planar graph of a given
minimum degree, then defer to ``dual_cut_random_face``.
* ``dual_cut_random_face(G, ...)`` -- choose a random face, deep-embed
relative to it, and use the cap vertex as the source, then defer to
``dual_cut_random_entry``.
* ``dual_cut_random_entry(G', cap, ...)`` -- choose a random root entry
tooth, then defer to ``medial_tire_dual_cut``.
* ``medial_tire_dual_cut(G', source, entry_edge)`` -- the worker: chain the
walk-depth labelling/cut from the given root entry tooth and assemble the
source-dual cut.
Run with the repo venv (networkx; matplotlib only for ``--png``):
``.venv/bin/python``.
"""
from __future__ import annotations
import argparse
import os
import random
import sys
from collections import defaultdict
import networkx as nx
_HERE = os.path.dirname(os.path.abspath(__file__))
_MTD = os.path.normpath(os.path.join(
_HERE, "..", "..",
"medial_tire_decompositions_of_plane_triangulations", "experiments"))
sys.path.insert(0, _MTD)
sys.path.insert(0, _HERE)
from tire_realization_analysis import ( # noqa: E402
ekey, extract_tread, medial_graph, medial_tire_facemodel,
recognise, triangular_faces,
)
from run_medial_tire_cut_experiment import ( # noqa: E402
_assemble_cut_graph, _cap_cut, _label_treads,
random_maximal_planar_min_degree,
)
# --------------------------------------------------------------------------- #
# Tread recognition and the source-dual graph.
# --------------------------------------------------------------------------- #
def _build_treads(faces, levels):
"""Recognise the full medial tire graph(s) of every BFS-level tread.
A tread depth whose annular frontier splits into several disjoint cycles
yields one tire per cycle. Returns ``(treads, skipped)`` where ``treads``
maps ``(depth, component)`` to the recognised ``(g, bij)`` and ``skipped``
lists ``(d, reason)`` for the depths that produced no tire.
"""
treads, skipped = {}, []
for d in range(max(levels.values())):
tread = extract_tread(faces, levels, d)
if tread is None:
skipped.append((d, "no tread faces"))
continue
if len(tread["up"]) < 3:
skipped.append((d, f"only {len(tread['up'])} up teeth"))
continue
tires = recognise(medial_tire_facemodel(tread["tread_faces"]), tread)
if not tires:
skipped.append((d, "no annular cycle recognised as a tire"))
continue
for c, gb in enumerate(tires):
treads[(d, c)] = gb
return treads, skipped
def root_entry_choices(G, source):
"""Edge indices of the root tread's up teeth -- the eligible entry teeth.
Empty when ``source`` induces no recognised root tread.
"""
faces, _ = triangular_faces(G)
levels = nx.single_source_shortest_path_length(G, source)
treads, _ = _build_treads(faces, levels)
if not treads:
return []
g, _bij = treads[min(treads)]
return sorted(g.up_edges)
# --------------------------------------------------------------------------- #
# Deep embedding (relative to a chosen face) and its outer-cap vertex.
# --------------------------------------------------------------------------- #
def _plane_depth(G, outer_cycle):
"""Plane depth of every vertex of ``G`` relative to ``outer_cycle``: the
graph distance to the nearest outer-cycle vertex (outer cycle = depth 0).
Mirrors ``plane_depth_sequencing.get_plane_depth_labelling`` -- attach a
temporary super-source to the outer cycle, BFS, and subtract one."""
tmp = G.copy()
s = max(G.nodes()) + 1
tmp.add_node(s)
tmp.add_edges_from((s, v) for v in outer_cycle)
dist = nx.single_source_shortest_path_length(tmp, s)
return {v: dist[v] - 1 for v in G.nodes()}
def deep_embedding(G, face):
"""Deep embedding of maximal planar ``G`` relative to triangular ``face``.
Networkx port of ``plane_depth_sequencing.extended_deep_embedding``: with
``face`` taken as the outer face, subdivide every *neutral* triangular face
(all three vertices at equal plane depth) -- including ``face`` itself -- by
inserting a new vertex adjacent to its three corners. The vertex inserted
inside ``face`` is the outer-cap vertex x* (depth -1); the rest sit one
level deeper than the face they cap.
Returns ``(G_prime, cap_vertex, depth)``.
"""
faces, _ = triangular_faces(G)
outer = frozenset(face)
depth = _plane_depth(G, face)
G_prime = G.copy()
nxt = max(G.nodes()) + 1
cap_vertex = None
for f in faces:
assert len(f) == 3, f"non-triangular face {f} (graph not maximal planar?)"
a, b, c = f
if depth[a] == depth[b] == depth[c]:
x = nxt
nxt += 1
G_prime.add_node(x)
G_prime.add_edges_from([(x, a), (x, b), (x, c)])
if frozenset(f) == outer:
cap_vertex = x
depth[x] = -1
else:
depth[x] = depth[a] + 1
if cap_vertex is None:
raise ValueError(f"face {face} is not a face of G")
return G_prime, cap_vertex, depth
def deep_embed_random_face(G, rng=None):
"""Pick a random triangular face of ``G`` and deep-embed relative to it.
Returns ``(G_prime, cap_vertex, face)``; ``cap_vertex`` is the outer-cap
vertex used as the level source."""
rng = rng or random.Random()
faces, _ = triangular_faces(G)
face = rng.choice(faces)
G_prime, cap, _depth = deep_embedding(G, face)
return G_prime, cap, face
def source_dual(G, faces):
"""The planar dual of triangulation ``G``: one node per face, one edge per
primal edge (tagged ``primal``). Faces are indexed as in ``faces``."""
edge_faces = defaultdict(list)
for fi, f in enumerate(faces):
for a, b in ((f[0], f[1]), (f[1], f[2]), (f[2], f[0])):
edge_faces[ekey(a, b)].append(fi)
D = nx.Graph()
D.add_nodes_from(range(len(faces)))
for e, fs in edge_faces.items():
if len(fs) == 2:
D.add_edge(fs[0], fs[1], primal=e)
return D
def annular_cut_edges(results, cap_cuts):
"""Primal edges whose dual edge a *closing* cut removes: the cap cut plus
each tread's annular-vertex duplications."""
removed = set()
for c in cap_cuts or []:
removed.add(c["medial_vertex"])
for key in sorted(results):
bij = results[key]["bij"]
for c in results[key]["cuts"]:
if c.vertex is not None:
removed.add(bij[f"a{c.vertex}"])
return removed
def up_apex_cut_edges(results):
"""Primal edges whose dual edge the apex duplications remove: the apex
medial vertex of every (singleton) up tooth across all treads, except the
entry tooth of each tread (its apex is not duplicated)."""
removed = set()
for key in sorted(results):
g, bij = results[key]["g"], results[key]["bij"]
entry = results[key]["entry_edge"]
for i in g.up_edges:
if i == entry:
continue
removed.add(bij[f"u{i}"])
return removed
def removed_dual_edges(results, cap_cuts):
"""All primal edges whose dual edge the cut removes: the closing annular
cuts together with the up-tooth apex duplications."""
return annular_cut_edges(results, cap_cuts) | up_apex_cut_edges(results)
def dual_face_missing(G, removed):
"""For each dual face (vertex ``v`` of ``G``), the number of bounding dual
edges removed by the cut."""
return {v: sum(1 for w in G.neighbors(v) if ekey(v, w) in removed)
for v in G.nodes()}
# --------------------------------------------------------------------------- #
# The four chained entry points.
# --------------------------------------------------------------------------- #
def medial_tire_dual_cut(G, source, entry_edge):
"""Chain the walk-depth labelling/cut from root entry tooth ``entry_edge``
and assemble the source-dual cut of ``G`` at level ``source``.
``entry_edge`` must be an up tooth of the root (lowest recognised) tread;
see ``root_entry_choices``. Returns a structured result dict.
"""
faces, emb = triangular_faces(G)
M = medial_graph(G)
levels = nx.single_source_shortest_path_length(G, source)
treads, skipped = _build_treads(faces, levels)
if not treads:
raise ValueError(f"level source {source} induces no recognised tread")
g_root = treads[min(treads)][0]
if entry_edge not in g_root.up_edges:
raise ValueError(
f"entry edge {entry_edge} is not an up tooth of the root tread "
f"(choices: {sorted(g_root.up_edges)})")
results = {}
_label_treads(treads, results, root_entry_edge=entry_edge)
cap_cuts = _cap_cut(G, emb, source, levels, results)
cut_graph, labels, warnings = _assemble_cut_graph(M, results, cap_cuts=cap_cuts)
dual = source_dual(G, faces)
annular = annular_cut_edges(results, cap_cuts)
apex = up_apex_cut_edges(results)
removed = annular | apex
missing = dual_face_missing(G, removed)
# The first entry: the medial vertex (primal edge) of the root tread's
# entry up-tooth apex. This apex is *not* duplicated, so it is the seam the
# chained walk starts from rather than a removed edge.
root = min(results)
entry_medial = results[root]["bij"][f"u{entry_edge}"]
return {
"G": G, "M": M, "source": source, "entry_edge": entry_edge,
"entry_medial_vertex": entry_medial,
"faces": faces, "outer_face": 0,
"levels": levels, "treads": treads, "skipped": skipped,
"results": results, "cap_cuts": cap_cuts, "cut_graph": cut_graph,
"labels": labels, "warnings": warnings,
"dual": dual, "removed_dual_edges": removed,
"annular_cut_edges": annular, "apex_cut_edges": apex,
"dual_face_missing": missing,
"max_missing": max(missing.values()) if missing else 0,
}
def dual_cut_random_entry(G, source, rng=None):
"""Pick a random root entry tooth at ``source``, then ``medial_tire_dual_cut``."""
rng = rng or random.Random()
choices = root_entry_choices(G, source)
if not choices:
raise ValueError(f"level source {source} induces no recognised root tread")
return medial_tire_dual_cut(G, source, rng.choice(choices))
def dual_cut_random_face(G, rng=None):
"""Pick a random face of ``G``, deep-embed relative to it, and cut from the
resulting outer-cap vertex, then ``dual_cut_random_entry``.
Faces are tried in random order; the first whose cap vertex induces a
recognised root tread is used. The dual cut is then read off the deep
embedding ``G'`` (stored as ``result["G"]``); the original triangulation is
kept as ``result["base_graph"]``."""
rng = rng or random.Random()
faces, _ = triangular_faces(G)
order = list(faces)
rng.shuffle(order)
for face in order:
G_prime, cap, depth = deep_embedding(G, face)
if root_entry_choices(G_prime, cap):
result = dual_cut_random_entry(G_prime, cap, rng=rng)
result["base_graph"] = G
result["chosen_face"] = tuple(face)
result["cap_vertex"] = cap
result["deep_depth"] = depth
return result
raise ValueError("no face's cap vertex induces a recognised root tread")
def random_dual_cut(n=20, seed=0, rng=None, min_degree=5, flips=400, attempts=1000):
"""Find a random maximal planar graph of minimum degree ``min_degree``, then
``dual_cut_random_face``.
``seed`` drives the graph sample; ``rng`` (defaulting to ``Random(seed)``)
drives the random face, deep embedding, and entry choices, so the whole
pipeline is reproducible from ``(n, seed)``.
"""
rng = rng or random.Random(seed)
G, graph_seed = random_maximal_planar_min_degree(
n, seed, flips=flips, min_degree=min_degree, attempts=attempts)
result = dual_cut_random_face(G, rng=rng)
result["graph_seed"] = graph_seed
result["base_min_degree"] = min(dict(G.degree()).values())
result["min_degree"] = min(dict(result["G"].degree()).values())
return result
# --------------------------------------------------------------------------- #
# Reporting and (optional) rendering.
# --------------------------------------------------------------------------- #
def summary(result):
G, missing = result["G"], result["dual_face_missing"]
removed = result["removed_dual_edges"]
hist = defaultdict(int)
for k in missing.values():
hist[k] += 1
base = result.get("base_graph")
lines = [
f"source-dual cut: n={G.number_of_nodes()} "
f"(deep embedding of base n="
f"{base.number_of_nodes() if base is not None else '?'}) "
f"graph_seed={result.get('graph_seed', '?')} "
f"min_degree={result.get('min_degree', min(dict(G.degree()).values()))}",
f"chosen face: {result.get('chosen_face', '?')} "
f"-> cap vertex x*={result.get('cap_vertex', result['source'])}",
f"level source: cap vertex {result['source']} "
f"root entry tooth: e{result['entry_edge']}",
f"recognised tires (depth.component): "
f"{[f'{d}.{c}' for d, c in sorted(result['treads'])]} "
f"skipped: {result['skipped']}",
f"removed source-dual edges ({len(removed)}): "
f"{len(result['annular_cut_edges'])} annular/cap + "
f"{len(result['apex_cut_edges'])} up-tooth apex",
f" annular/cap: {sorted(result['annular_cut_edges'])}",
f" up apexes: {sorted(result['apex_cut_edges'])}",
f"dual-face missing-edge histogram (count by #removed around the dual "
f"face): {dict(sorted(hist.items()))} max={result['max_missing']}",
]
for v in sorted(missing, key=lambda v: (-missing[v], v)):
if missing[v]:
inc = [ekey(v, w) for w in G.neighbors(v) if ekey(v, w) in removed]
lines.append(f" dual face v{v} (deg {G.degree(v)}): "
f"{missing[v]} missing -> {inc}")
return "\n".join(lines)
def _radial_source_layout(G, source, levels):
"""Concentric ('onion') layout rooted at the cap ``source``: radius grows
with BFS level so the depth rings are actual circles, and each ring's
angular order is inherited from its lower-level neighbours to keep the
nesting legible. This matches the cap-source construction, where the BFS
rings are exactly the plane-depth rings."""
import math
max_level = max(levels.values()) or 1
ring = defaultdict(list)
for v, d in levels.items():
ring[d].append(v)
angle = {source: 0.0}
pos = {source: (0.0, 0.0)}
for d in range(1, max_level + 1):
verts = ring[d]
prov = {}
for v in verts:
pa = [angle[w] for w in G.neighbors(v)
if levels.get(w) == d - 1 and w in angle]
if pa:
sx = sum(math.cos(a) for a in pa)
sy = sum(math.sin(a) for a in pa)
prov[v] = math.atan2(sy, sx)
else:
prov[v] = 0.0
verts.sort(key=lambda v: prov[v] % (2 * math.pi))
k = len(verts)
base = prov[verts[0]] if verts else 0.0
r = d / max_level
for i, v in enumerate(verts):
a = base + 2 * math.pi * i / k
angle[v] = a
pos[v] = (r * math.cos(a), r * math.sin(a))
return pos
def draw_png(result, path, scale=6.0):
"""Render the source-dual cut: dual nodes at face centroids, dual edges
drawn light gray where the cut removed them, labelled by missing count.
The source graph is laid out concentrically around the cap source so the
BFS/plane-depth rings read as nested circles."""
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
G, faces, dual = result["G"], result["faces"], result["dual"]
removed = result["removed_dual_edges"]
missing = result["dual_face_missing"]
source = result["source"]
entry_medial = result.get("entry_medial_vertex")
pos_v = _radial_source_layout(G, source, result["levels"])
def centroid(fi):
xs = [pos_v[u][0] for u in faces[fi]]
ys = [pos_v[u][1] for u in faces[fi]]
return (sum(xs) / 3.0, sum(ys) / 3.0)
pos = {fi: centroid(fi) for fi in dual.nodes()}
fig, ax = plt.subplots(figsize=(7.6, 7.6))
# primal (source) graph, faint, for orientation
for u, v in G.edges():
ax.plot([pos_v[u][0], pos_v[v][0]], [pos_v[u][1], pos_v[v][1]],
color="0.85", lw=0.5, zorder=0)
# the entry medial vertex = a primal edge; highlight that primal edge and
# the dual edge crossing it.
if entry_medial is not None:
eu, ev = entry_medial
ax.plot([pos_v[eu][0], pos_v[ev][0]], [pos_v[eu][1], pos_v[ev][1]],
color="#1b9e44", lw=2.6, zorder=2, solid_capstyle="round")
for u, v, data in dual.edges(data=True):
cut = data["primal"] in removed
is_entry = entry_medial is not None and data["primal"] == entry_medial
if is_entry:
ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
color="#1b9e44", lw=2.6, zorder=4, solid_capstyle="round")
mx, my = (pos[u][0] + pos[v][0]) / 2, (pos[u][1] + pos[v][1]) / 2
ax.plot(mx, my, "*", ms=15, mfc="#1b9e44", mec="white",
mew=0.7, zorder=5)
ax.text(mx, my - 0.06, "entry", color="#1b9e44", fontsize=8,
fontweight="bold", ha="center", va="top", zorder=5)
else:
ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
color="0.80" if cut else "0.25",
lw=1.0 if cut else 1.3,
linestyle=(0, (2, 2)) if cut else "solid", zorder=1)
for fi in dual.nodes():
x, y = pos[fi]
ax.plot(x, y, "o", ms=4, color="#3a6ea5", zorder=3)
# label each source-graph vertex by its id; the cap source is flagged.
for v in G.nodes():
x, y = pos_v[v]
is_src = v == source
ax.text(x, y, str(v),
color="#0b6", fontsize=8 if not is_src else 9,
fontweight="bold" if is_src else "normal",
ha="center", va="center", zorder=6,
bbox=dict(boxstyle="round,pad=0.12",
fc="#eafff2" if is_src else "white",
ec="#1b9e44" if is_src else "0.6", lw=0.7))
# missing-edge count, offset above-right of the vertex label.
m = missing[v]
if m:
ax.text(x + 0.045, y + 0.045, str(m), color="#b03030", fontsize=7,
ha="left", va="bottom", zorder=7,
bbox=dict(boxstyle="circle,pad=0.05", fc="white",
ec="#b03030", lw=0.6))
ax.set_title(f"source-dual cut (cap source {source}, entry "
f"e{result['entry_edge']} = medial vtx {entry_medial}); "
f"gray = edges missing after cuts\n"
f"green star = first entry medial vertex; red numbers = "
f"#missing dual edges around each dual face; "
f"max {result['max_missing']}", fontsize=9)
ax.set_aspect("equal")
ax.axis("off")
fig.tight_layout()
fig.savefig(path, dpi=150)
plt.close(fig)
def _tire_coords(g, r_ann=1.0, r_up=1.46, r_down=0.60):
"""Annular/teeth coordinates for one tread, matching
``medial_tire_cut_labelling.to_tikz``: a_0 at the top, k increasing CW."""
import math
n = g.n
def ang(k):
return math.radians(90.0 - k * 360.0 / n)
def mid(i):
a0, a1 = ang(i), ang((i + 1) % n)
return math.atan2(math.sin(a0) + math.sin(a1), math.cos(a0) + math.cos(a1))
pos = {f"a{k}": (r_ann * math.cos(ang(k)), r_ann * math.sin(ang(k)))
for k in range(n)}
for i in g.up_edges:
a = mid(i)
pos[f"u{i}"] = (r_up * math.cos(a), r_up * math.sin(a))
for i in g.singleton_down_edges:
a = mid(i)
pos[f"d{i}"] = (r_down * math.cos(a), r_down * math.sin(a))
for (i, j) in g.bites:
pts = [pos[f"a{i}"], pos[f"a{(i + 1) % n}"],
pos[f"a{j}"], pos[f"a{(j + 1) % n}"]]
cx = sum(p[0] for p in pts) / 4.0
cy = sum(p[1] for p in pts) / 4.0
pos[f"p{i}_{j}"] = (0.9 * cx, 0.9 * cy)
return pos
def _draw_tread(ax, g, depth, cuts, entry_edge, title):
"""Draw one full medial tire cut on ``ax`` (annular cycle, teeth, walk-depth
labels, cut slits), mirroring ``medial_tire_cut_labelling.to_tikz``."""
import math
n = g.n
pos = _tire_coords(g)
def seg(a, b, **kw):
ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]], **kw)
# annular cycle
xs = [pos[f"a{k}"][0] for k in range(n)] + [pos["a0"][0]]
ys = [pos[f"a{k}"][1] for k in range(n)] + [pos["a0"][1]]
ax.plot(xs, ys, color="black", lw=1.4, zorder=1)
# spokes (teeth)
for i in g.up_edges:
seg(f"u{i}", f"a{i}", color="0.55", lw=0.6, zorder=1)
seg(f"u{i}", f"a{(i + 1) % n}", color="0.55", lw=0.6, zorder=1)
for i in g.singleton_down_edges:
seg(f"d{i}", f"a{i}", color="0.55", lw=0.6, zorder=1)
seg(f"d{i}", f"a{(i + 1) % n}", color="0.55", lw=0.6, zorder=1)
for (i, j) in g.bites:
apex = f"p{i}_{j}"
for e in (i, j):
seg(apex, f"a{e}", color="0.55", lw=0.6, zorder=1)
seg(apex, f"a{(e + 1) % n}", color="0.55", lw=0.6, zorder=1)
# vertices
for k in range(n):
ax.plot(*pos[f"a{k}"], "o", ms=3, color="black", zorder=3)
for i in g.up_edges:
ax.plot(*pos[f"u{i}"], "o", ms=5, mfc="#cfe0f3", mec="#3a6ea5", zorder=3)
for i in g.singleton_down_edges:
ax.plot(*pos[f"d{i}"], "o", ms=5, mfc="#f3cfcf", mec="#a53a3a", zorder=3)
for (i, j) in g.bites:
ax.plot(*pos[f"p{i}_{j}"], "o", ms=6, mfc="#e8a0a0", mec="#a53a3a", zorder=3)
# walk-depth labels along each spoke
if depth is not None:
for edge in range(n):
ax_, ay = pos[g.apex_of_edge(edge)]
a, b = pos[f"a{edge}"], pos[f"a{(edge + 1) % n}"]
mx, my = 0.5 * (a[0] + b[0]), 0.5 * (a[1] + b[1])
lx, ly = ax_ + 0.5 * (mx - ax_), ay + 0.5 * (my - ay)
ax.text(lx, ly, str(depth[edge]), fontsize=7, fontweight="bold",
ha="center", va="center", zorder=4)
# annular-vertex cut slits (radial)
for c in cuts or []:
if c.vertex is None:
continue
vx, vy = pos[f"a{c.vertex}"]
rad = math.atan2(vy, vx)
dx, dy = 0.16 * math.cos(rad), 0.16 * math.sin(rad)
ax.plot([vx - dx, vx + dx], [vy - dy, vy + dy],
color="#cc2020", lw=2.0, zorder=5)
ax.text(vx + 0.34 * math.cos(rad), vy + 0.34 * math.sin(rad),
f"cut {c.order + 1}", fontsize=6, color="#cc2020",
ha="center", va="center", zorder=5)
# up-tooth apex duplications (slit tangential, across the apex marker);
# the entry tooth's apex is not duplicated
for i in g.up_edges:
if i == entry_edge:
continue
vx, vy = pos[f"u{i}"]
rad = math.atan2(vy, vx)
tx, ty = -math.sin(rad), math.cos(rad) # tangential
ax.plot([vx - 0.12 * tx, vx + 0.12 * tx],
[vy - 0.12 * ty, vy + 0.12 * ty],
color="#cc2020", lw=2.0, zorder=6)
# entry marker
if entry_edge is not None:
ex, ey = pos[g.apex_of_edge(entry_edge)]
rad = math.atan2(ey, ex)
ax.text(ex + 0.34 * math.cos(rad), ey + 0.34 * math.sin(rad),
"entry", fontsize=6, color="#3a6ea5", ha="center", va="center")
ax.set_title(title, fontsize=8)
ax.set_aspect("equal")
ax.axis("off")
def draw_tire_cuts_png(result, path):
"""Render every recognised tire's full medial tire cut, one panel each.
A tread depth with several disjoint annular cycles contributes one panel
per cycle, labelled ``tread d.c``."""
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
res = result["results"]
keys = sorted(res)
if not keys:
raise ValueError("no recognised tires to draw")
fig, axes = plt.subplots(1, len(keys), figsize=(5.2 * len(keys), 5.4))
if len(keys) == 1:
axes = [axes]
for ax, key in zip(axes, keys):
d, comp = key
rec = res[key]
g = rec["g"]
title = (f"tread {d}.{comp}: |A(T)|={g.n} word={g.tooth_word}\n"
f"bites={sorted(g.bites)} entry=e{rec['entry_edge']} "
f"start_depth={rec['start_depth']} cuts={len(rec['cuts'])}")
_draw_tread(ax, g, rec["depth"], rec["cuts"], rec["entry_edge"], title)
fig.suptitle(f"full medial tire cuts -- source {result['source']}, "
f"root entry e{result['entry_edge']}", fontsize=10)
fig.tight_layout()
fig.savefig(path, dpi=150)
plt.close(fig)
def draw_cap_png(result, path):
"""Render tread 0, the source cap: a wheel with the source at the hub, its
link cycle as the rim, the cap triangles (down teeth) filled, and the cap
cut marked. Tread 0 is skipped by tire recognition (a wheel has no up
teeth), so this draws the ``extract_tread`` roles directly."""
import math
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
G, source = result["G"], result["source"]
faces, emb = triangular_faces(G)
levels = nx.single_source_shortest_path_length(G, source)
tr = extract_tread(faces, levels, 0)
if tr is None:
raise ValueError("no tread-0 (cap) faces")
link = list(emb.neighbors_cw_order(source))
cap_cuts = {c["medial_vertex"] for c in result.get("cap_cuts", [])}
pos = {source: (0.0, 0.0)}
k = len(link)
for i, v in enumerate(link):
a = math.radians(90 - i * 360.0 / k)
pos[v] = (math.cos(a), math.sin(a))
fig, ax = plt.subplots(figsize=(6.5, 6.8))
for f in tr["tread_faces"]:
if all(v in pos for v in f):
xy = [pos[v] for v in f]
ax.fill([p[0] for p in xy], [p[1] for p in xy],
color="#eef3fa", zorder=0)
def edge(u, v, **kw):
ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]], **kw)
for u, v in tr["annular"]: # spokes (source -> link)
edge(u, v, color="0.45", lw=1.0, zorder=1)
for u, v in tr["down"]: # link cycle (down-tooth bases)
edge(u, v, color="black", lw=1.6, zorder=1)
ax.plot(*pos[source], "o", ms=11, mfc="#cfe0f3", mec="#3a6ea5", zorder=4)
ax.text(*pos[source], str(source), ha="center", va="center", fontsize=9,
fontweight="bold", color="#234", zorder=5)
for v in link:
ax.plot(*pos[v], "o", ms=9, mfc="white", mec="black", zorder=4)
x, y = pos[v]
ax.text(x * 1.13, y * 1.13, str(v), ha="center", va="center", fontsize=9)
for u, v in list(tr["annular"]) + list(tr["down"]):
mx, my = (pos[u][0] + pos[v][0]) / 2, (pos[u][1] + pos[v][1]) / 2
cut = ekey(u, v) in cap_cuts
ax.plot(mx, my, "s", ms=5, mfc=("#cc2020" if cut else "#888"),
mec="none", zorder=3)
if cut:
dx, dy = pos[v][0] - pos[u][0], pos[v][1] - pos[u][1]
L = math.hypot(dx, dy) or 1.0
px, py = -dy / L * 0.13, dx / L * 0.13
ax.plot([mx - px, mx + px], [my - py, my + py],
color="#cc2020", lw=2.2, zorder=5)
ax.text(mx + 0.12, my, "cap cut", color="#cc2020", fontsize=7,
va="center")
ax.set_title(f"tread 0 (source cap) -- source {source}, link {link}\n"
f"{len(tr['tread_faces'])} cap triangles; no up teeth (skipped); "
f"down teeth = link cycle", fontsize=8)
ax.set_aspect("equal")
ax.axis("off")
ax.set_xlim(-1.4, 1.4)
ax.set_ylim(-1.4, 1.4)
fig.tight_layout()
fig.savefig(path, dpi=150)
plt.close(fig)
def main():
parser = argparse.ArgumentParser(
description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("-n", type=int, default=20, help="number of vertices")
parser.add_argument("--seed", type=int, default=0, help="graph sample seed")
parser.add_argument("--min-degree", type=int, default=5)
parser.add_argument("--face", type=str, default=None,
help="fix the chosen face as 'a,b,c' (default: random "
"via rng); the deep embedding's cap vertex is the "
"source")
parser.add_argument("--entry", type=int, default=None,
help="fix the root entry tooth (requires --face)")
parser.add_argument("--png", metavar="PATH", help="render the dual cut to PNG")
parser.add_argument("--tire-png", metavar="PATH",
help="render each full medial tire cut to PNG")
parser.add_argument("--cap-png", metavar="PATH",
help="render tread 0 (the source cap) to PNG")
args = parser.parse_args()
rng = random.Random(args.seed)
if args.face is not None:
G, graph_seed = random_maximal_planar_min_degree(
args.n, args.seed, min_degree=args.min_degree)
face = tuple(int(x) for x in args.face.split(","))
G_prime, cap, depth = deep_embedding(G, face)
if args.entry is not None:
result = medial_tire_dual_cut(G_prime, cap, args.entry)
else:
result = dual_cut_random_entry(G_prime, cap, rng=rng)
result["base_graph"] = G
result["chosen_face"] = face
result["cap_vertex"] = cap
result["deep_depth"] = depth
result["graph_seed"] = graph_seed
result["base_min_degree"] = min(dict(G.degree()).values())
result["min_degree"] = min(dict(G_prime.degree()).values())
else:
result = random_dual_cut(n=args.n, seed=args.seed,
rng=rng, min_degree=args.min_degree)
print(summary(result))
if args.png:
draw_png(result, args.png)
print(f"wrote {args.png}")
if args.tire_png:
draw_tire_cuts_png(result, args.tire_png)
print(f"wrote {args.tire_png}")
if args.cap_png:
draw_cap_png(result, args.cap_png)
print(f"wrote {args.cap_png}")
if __name__ == "__main__":
main()
papers/medial_tire_cuts/experiments/run_medial_tire_cut_experiment.py
-445
View File
@@ -1,445 +0,0 @@
"""Medial tire cut experiment.
End-to-end experiment for the *Medial Tire Cuts* paper:
1. Generate a random maximal planar graph G on n vertices (stacked seed plus
random diagonal flips; ``random_maximal_planar`` from the medial tire
decompositions experiments), optionally rejecting samples below a requested
minimum degree.
2. Build its medial graph M(G).
3. Take the nested tire decomposition at one random vertex level source: the
BFS-level treads, each realized as a FullMedialTireGraph.
4. Walk-depth label and cut each full medial tire graph, chaining the labels
down the tire tree, and assemble one final cut graph of M(G) with a global
label map.
This script produces *data* (the final cut graph and its labels); it draws
nothing. Anything for the paper (figures) lives in a separate script that
imports ``run_experiment`` from here.
Chaining rule (walk depths across the tire tree).
* The root tread (no recognised parent) is entered at an arbitrary up tooth
with walk depth 0.
* A child tread is entered at the up tooth whose apex is the *same medial
vertex* as the parent's down tooth of lowest walk depth -- a parent down
tooth and the child up tooth glued to it across the shared level cycle are
the same medial vertex of M(G). The entry up tooth's walk depth is that
parent down tooth's depth + 1, and the walk increments locally from there.
* The source cap contributes one additional cut. It is placed at the
counter-clockwise source edge incident to the cap down tooth whose apex is
the root tread's entry up tooth.
Run with the repo venv (networkx + scipy): ``.venv/bin/python``.
"""
from __future__ import annotations
import argparse
import json
import os
import random
import subprocess
import sys
import networkx as nx
# Reuse the realization pipeline from the medial tire decompositions paper, and
# the walk-depth labelling/cut from this paper's companion script.
_HERE = os.path.dirname(os.path.abspath(__file__))
_MTD = os.path.normpath(os.path.join(
_HERE, "..", "..",
"medial_tire_decompositions_of_plane_triangulations", "experiments"))
sys.path.insert(0, _MTD)
sys.path.insert(0, _HERE)
from tire_realization_analysis import ( # noqa: E402
ekey, extract_tread, medial_graph, medial_tire_facemodel,
random_maximal_planar, recognise, triangular_faces,
)
from medial_tire_cut_labelling import door_bite, label_and_cut # noqa: E402
# --------------------------------------------------------------------------- #
# 4. Walk-depth labelling and cut, chained down the tire tree.
# --------------------------------------------------------------------------- #
def _apex_vertex(g, bij, edge):
"""The medial vertex that is the apex of the tooth on annular ``edge``."""
return bij[g.apex_of_edge(edge)]
def _label_treads(treads, results, root_entry_edge=None):
"""Fill ``results[(d, c)]`` with the walk-depth labelling and cuts for every
recognised tire ``c`` of every tread depth ``d``, chaining child entries to
parent down teeth.
``treads`` maps ``(depth, component)`` -> ``(g, bij)``; a tread depth may
carry several tires (one per disjoint annular cycle). The root tire
``(root_d, 0)`` is entered at ``root_entry_edge`` when given -- it must be
one of that tire's up teeth -- otherwise at an arbitrary up tooth. Each
other tire chains to whichever parent-depth down tooth (across all parent
tires) shares its apex, at the lowest parent walk depth.
"""
if not treads:
return
depths = sorted({k[0] for k in treads})
root_d = depths[0]
for d in depths:
# apex medial vertex -> child start depth, over all parent-depth tires
parent_down = {}
for pk in (k for k in treads if k[0] == d - 1):
pg, pbij = treads[pk]
pdepth = results[pk]["depth"]
for e in pg.down_edges:
apex = _apex_vertex(pg, pbij, e)
value = pdepth[e] + 1
if apex not in parent_down or value < parent_down[apex]:
parent_down[apex] = value
has_parent = any(k[0] == d - 1 for k in treads)
for key in sorted(k for k in treads if k[0] == d):
g, bij = treads[key]
if not has_parent:
if (key == (root_d, 0) and root_entry_edge is not None
and root_entry_edge in g.up_edges):
entry_edge, start_depth = root_entry_edge, 0
else:
entry_edge, start_depth = g.up_edges[0], 0 # arbitrary entry
else:
child_up_apex = {bij[f"u{m}"]: m for m in g.up_edges}
best = None
for apex, value in parent_down.items():
if apex in child_up_apex and (best is None or value < best[1]):
best = (child_up_apex[apex], value)
if best is not None: # chains to a parent down tooth
entry_edge, start_depth = best
else: # no shared apex (degenerate); root-style
entry_edge, start_depth = g.up_edges[0], 0
depth, cuts = label_and_cut(g, entry_edge, start_depth=start_depth)
results[key] = {"g": g, "bij": bij, "entry_edge": entry_edge,
"start_depth": start_depth, "depth": depth,
"cuts": cuts}
def _cap_cut(G, emb, source, levels, results):
"""The source-cap cut determined by the first recognised tread's entry.
If the root tread enters at an up tooth whose apex is the level-1 edge
``xy``, then ``xy`` is a down tooth of the source cap. Cut the
counter-clockwise source edge incident to that down tooth. The returned
record also stores the local neighbour split used to unzip the medial
vertex in the whole medial graph.
"""
if not results:
return []
root_depth = min(results)
rec = results[root_depth]
g, bij = rec["g"], rec["bij"]
x, y = _apex_vertex(g, bij, rec["entry_edge"])
if levels.get(x) != 1 or levels.get(y) != 1:
return []
order = list(emb.neighbors_cw_order(source))
if x not in order or y not in order:
return []
next_cw = {v: order[(i + 1) % len(order)] for i, v in enumerate(order)}
prev_cw = {v: order[(i - 1) % len(order)] for i, v in enumerate(order)}
if next_cw[x] == y:
cut_endpoint, other_endpoint = x, y
elif next_cw[y] == x:
cut_endpoint, other_endpoint = y, x
else:
return []
other_cap_endpoint = prev_cw[cut_endpoint]
mv = ekey(source, cut_endpoint)
return [{
"medial_vertex": mv,
"down_tooth": ekey(cut_endpoint, other_endpoint),
"neighbours_a": [
ekey(source, other_endpoint),
ekey(cut_endpoint, other_endpoint),
],
"neighbours_b": [
ekey(source, other_cap_endpoint),
ekey(cut_endpoint, other_cap_endpoint),
],
}]
# --------------------------------------------------------------------------- #
# Assemble one final cut graph of M(G) with a global label map.
# --------------------------------------------------------------------------- #
def _assemble_cut_graph(M, results, cap_cuts=None):
"""Apply every tread's cuts to M(G).
Each cut duplicates an annular medial vertex, splitting its four incident
medial edges along the slit between the two teeth meeting there: the tooth
on the previous annular edge (with that edge's far annular vertex) goes to
one copy, the tooth on the next annular edge to the other.
Returns ``(H, label_records, warnings)`` where ``H`` is the cut graph (a
networkx graph whose split vertices are keyed ``(medial_vertex, "A"/"B",
tread)``) and ``label_records`` lists every tooth's walk depth.
"""
# Per cut annular vertex: map each original neighbour -> which copy keeps it.
split = {} # medial_vertex -> {neighbour_medial_vertex: copy_node}
warnings = []
for i, c in enumerate(cap_cuts or []):
mv = c["medial_vertex"]
if mv in split:
warnings.append(f"cap cut at {mv} was already cut; skipped")
continue
copy_a = (mv, "A", f"cap{i}")
copy_b = (mv, "B", f"cap{i}")
split[mv] = {
**{v: copy_a for v in c["neighbours_a"]},
**{v: copy_b for v in c["neighbours_b"]},
}
for key in sorted(results):
td = key[0]
g, bij = results[key]["g"], results[key]["bij"]
n = g.n
for c in results[key]["cuts"]:
kk = c.vertex
if kk is None:
continue
mv = bij[f"a{kk}"]
if mv in split:
warnings.append(f"annular vertex a{kk} of tread {key} cut twice; "
f"second cut not applied")
continue
e_prev, e_next = (kk - 1) % n, kk
copy_a = (mv, "A", td)
copy_b = (mv, "B", td)
split[mv] = {
bij[f"a{(kk - 1) % n}"]: copy_a,
_apex_vertex(g, bij, e_prev): copy_a,
bij[f"a{(kk + 1) % n}"]: copy_b,
_apex_vertex(g, bij, e_next): copy_b,
}
def resolve(node, other):
return split[node][other] if node in split else node
H = nx.Graph()
H.add_nodes_from(v for v in M.nodes() if v not in split)
for v, copies in split.items():
H.add_nodes_from(set(copies.values()))
for u, v in M.edges():
H.add_edge(resolve(u, v), resolve(v, u))
label_records = []
for key in sorted(results):
td = key[0]
g, bij, depth = results[key]["g"], results[key]["bij"], results[key]["depth"]
for k in range(g.n):
role = ("up" if g.tooth_word[k] == "U"
else "bite" if door_bite(g, k) is not None else "down")
label_records.append({
"tread": td, "comp": key[1], "edge": k, "role": role,
"apex": _apex_vertex(g, bij, k), "walk": depth[k],
})
return H, label_records, warnings
# --------------------------------------------------------------------------- #
# Driver.
# --------------------------------------------------------------------------- #
def random_maximal_planar_min_degree(n: int, seed: int, flips: int = 400,
min_degree: int = 0,
attempts: int = 1000) -> tuple[nx.Graph, int]:
"""Generate a random maximal planar graph with minimum degree at least
``min_degree``. The returned seed is the actual sample seed used."""
if min_degree <= 0:
return random_maximal_planar(n, seed, flips=flips), seed
if min_degree >= 5:
plantri = os.path.normpath(os.path.join(_HERE, "..", "..", "..",
"plantri", "plantri"))
if os.path.exists(plantri):
data = subprocess.check_output(
[plantri, f"-m{min_degree}", "-g", str(n)],
stderr=subprocess.DEVNULL)
graphs = [line for line in data.splitlines() if line]
if graphs:
G = nx.from_graph6_bytes(graphs[seed % len(graphs)])
return nx.convert_node_labels_to_integers(G), seed
for offset in range(attempts):
sample_seed = seed + offset
G = random_maximal_planar(n, sample_seed, flips=flips)
if min(dict(G.degree()).values()) >= min_degree:
return G, sample_seed
raise RuntimeError(
f"no random maximal planar graph on {n} vertices with "
f"minimum degree >= {min_degree} found in {attempts} attempts "
f"starting at seed {seed}")
def run_experiment(n: int = 12, seed: int = 0, flips: int = 400,
min_degree: int = 5, attempts: int = 1000) -> dict:
"""Run the full pipeline and return a structured result.
Result keys: ``n, seed, G, M, source, treads`` (dict depth -> (g, bij)),
``results`` (dict depth -> labelling/cut record), ``skipped`` (list of
(depth, reason)), ``cut_graph`` (networkx graph), ``labels`` (list of tooth
records), ``warnings``.
"""
G, graph_seed = random_maximal_planar_min_degree(
n, seed, flips=flips, min_degree=min_degree, attempts=attempts)
faces, emb = triangular_faces(G)
M = medial_graph(G)
source = random.Random(f"source-{graph_seed}").choice(sorted(G.nodes()))
levels = nx.single_source_shortest_path_length(G, source)
treads, skipped = {}, []
for d in range(max(levels.values())):
tread = extract_tread(faces, levels, d)
if tread is None:
skipped.append((d, "no tread faces"))
continue
if len(tread["up"]) < 3:
skipped.append((d, f"only {len(tread['up'])} up teeth"))
continue
tires = recognise(medial_tire_facemodel(tread["tread_faces"]), tread)
if not tires:
skipped.append((d, "no annular cycle recognised as a tire"))
continue
for c, gb in enumerate(tires):
treads[(d, c)] = gb
results = {}
_label_treads(treads, results)
cap_cuts = _cap_cut(G, emb, source, levels, results)
cut_graph, labels, warnings = _assemble_cut_graph(M, results, cap_cuts=cap_cuts)
return {
"n": n, "seed": seed, "graph_seed": graph_seed,
"min_degree": min(dict(G.degree()).values()),
"G": G, "M": M, "source": source,
"treads": treads, "results": results, "cap_cuts": cap_cuts,
"skipped": skipped,
"cut_graph": cut_graph, "labels": labels, "warnings": warnings,
}
# --------------------------------------------------------------------------- #
# Serialization / reporting.
# --------------------------------------------------------------------------- #
def _vname(v) -> str:
"""Stable string name for a medial vertex (an edge key) or a split node."""
if isinstance(v, tuple) and len(v) == 3 and v[1] in ("A", "B"):
mv, side, d = v
return f"{mv[0]}-{mv[1]}#{side}@T{d}"
return f"{v[0]}-{v[1]}"
def to_json(result: dict) -> dict:
res = result["results"]
treads_out = []
for key in sorted(res):
d, comp = key
g, bij = res[key]["g"], res[key]["bij"]
depth, cuts = res[key]["depth"], res[key]["cuts"]
teeth = [{
"edge": k,
"role": ("up" if g.tooth_word[k] == "U"
else "bite" if door_bite(g, k) is not None else "down"),
"apex": _vname(_apex_vertex(g, bij, k)),
"walk": depth[k],
} for k in range(g.n)]
treads_out.append({
"depth": d, "comp": comp, "n": g.n, "tooth_word": g.tooth_word,
"bites": sorted(list(b) for b in g.bites),
"entry_edge": res[key]["entry_edge"], "start_depth": res[key]["start_depth"],
"teeth": teeth,
"cuts": [{
"annular_index": c.vertex,
"annular_vertex": _vname(bij[f"a{c.vertex}"]),
"last_edge": c.last_tooth, "closing_edge": c.closing_tooth,
} for c in cuts],
})
H = result["cut_graph"]
return {
"n": result["n"], "seed": result["seed"],
"graph_seed": result["graph_seed"], "min_degree": result["min_degree"],
"source": result["source"],
"graph_edges": sorted([int(u), int(v)] for u, v in result["G"].edges()),
"medial_vertices": result["M"].number_of_nodes(),
"skipped": [[d, why] for d, why in result["skipped"]],
"cap_cuts": [{
"medial_vertex": _vname(c["medial_vertex"]),
"down_tooth": _vname(c["down_tooth"]),
} for c in result["cap_cuts"]],
"treads": treads_out,
"cut_graph": {
"nodes": sorted(_vname(v) for v in H.nodes()),
"edges": sorted([_vname(u), _vname(v)] for u, v in H.edges()),
},
"labels": [{
"tread": r["tread"], "comp": r.get("comp", 0),
"edge": r["edge"], "role": r["role"],
"apex": _vname(r["apex"]), "walk": r["walk"],
} for r in result["labels"]],
"warnings": result["warnings"],
}
def summary(result: dict) -> str:
H, res = result["cut_graph"], result["results"]
lines = [
f"random maximal planar graph: n={result['n']} requested seed={result['seed']} "
f"graph seed={result['graph_seed']} "
f"({result['G'].number_of_edges()} edges, min degree {result['min_degree']})",
f"medial graph M(G): {result['M'].number_of_nodes()} vertices",
f"level source: vertex {result['source']}",
f"recognised tires (depth, component): {sorted(res)}",
f"skipped treads: {result['skipped']}",
]
for key in sorted(res):
d, comp = key
g = res[key]["g"]
ncuts = len(res[key]["cuts"])
lines.append(
f" tread {d}.{comp}: |A(T)|={g.n} word={g.tooth_word} "
f"bites={sorted(g.bites)} entry=e{res[key]['entry_edge']} "
f"start_depth={res[key]['start_depth']} cuts={ncuts}")
lines.append(
f"final cut graph: {H.number_of_nodes()} vertices, "
f"{H.number_of_edges()} edges, "
f"{len(result['cap_cuts']) + sum(len(r['cuts']) for r in res.values())} cuts total")
if result["warnings"]:
lines.append("warnings: " + "; ".join(result["warnings"]))
return "\n".join(lines)
def main() -> None:
parser = argparse.ArgumentParser(
description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("-n", type=int, default=12, help="number of vertices")
parser.add_argument("--seed", type=int, default=0)
parser.add_argument("--flips", type=int, default=400,
help="number of random diagonal flips when building G")
parser.add_argument("--min-degree", type=int, default=5,
help="reject random graphs below this minimum degree")
parser.add_argument("--attempts", type=int, default=1000,
help="number of consecutive seeds to try for --min-degree")
parser.add_argument("--json", metavar="PATH",
help="write the full result as JSON to PATH")
args = parser.parse_args()
result = run_experiment(n=args.n, seed=args.seed, flips=args.flips,
min_degree=args.min_degree, attempts=args.attempts)
print(summary(result))
if args.json:
with open(args.json, "w") as fh:
json.dump(to_json(result), fh, indent=2)
print(f"wrote {args.json}")
if __name__ == "__main__":
main()
papers/medial_tire_cuts/funcD_seed7_dual_cut.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 126 KiB

papers/medial_tire_cuts/funcD_seed7_tire_cuts.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 140 KiB

papers/medial_tire_cuts/paper.aux
-29
View File
@@ -1,29 +0,0 @@
\relax
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Cutting a full medial tire graph}}{1}{}\protected@file@percent }
\newlabel{def:walk-depth-cut}{{2.1}{1}}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\newlabel{rem:closing-tooth}{{2.2}{2}}
\newlabel{ex:worked-cut}{{2.3}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Chaining across the tire tree}}{2}{}\protected@file@percent }
\citation{bauerfeld-medial-tire}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A full medial tire graph (left) and its walk-depth labelling and cut (right), from Example\nonbreakingspace 2.3\hbox {}. Black vertices are the annular medial vertices of the cycle $A(T)$; blue vertices are up-tooth apexes, red vertices are down-tooth apexes, and the larger red vertex is the shared apex of the bite on annular edges $0$ and $4$. On the right, each tooth carries its walk depth, and the two red slits mark the cuts: \emph {cut\nonbreakingspace 1} duplicates $a_5$ as the root-face traversal closes, and \emph {cut\nonbreakingspace 2} duplicates $a_1$ as the bite's inner-gap face closes. After the cuts the only bounded faces are the eight teeth.}}{3}{}\protected@file@percent }
\newlabel{fig:worked-cut}{{1}{3}}
\newlabel{rem:chaining-candidates}{{3.1}{3}}
\newlabel{ex:real-cut}{{3.2}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The recognised tread $T_2$ of the medial tire decomposition of a random maximal planar graph on $20$ vertices (Example\nonbreakingspace 3.2\hbox {}), with its walk-depth labelling and cut. Black vertices are the annular medial vertices of $A(T)$; blue vertices are up-tooth apexes and red vertices down-tooth apexes, the larger red vertex being the shared apex of the bite on annular edges $2$ and $5$. Each tooth carries its walk depth; the red slits are the two cuts.}}{4}{}\protected@file@percent }
\newlabel{fig:real-cut}{{2}{4}}
\bibcite{bauerfeld-medial-tire}{1}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The source graph $G$ and the whole medial graph $M(G)$ of the minimum-degree-$5$ maximal planar graph on $20$ vertices generated by \texttt {plantri -m5} at seed $59$. The source vertex $5$ is highlighted in the top panel. In the bottom panel, each medial vertex is placed at the midpoint of its corresponding source edge and labelled by that edge. Black vertices come from source edges between consecutive levels; coloured vertices come from source edges within a single level of the chain. The red-highlighted vertices, walk-depth labels, and seven red slits are the computed source-cap cut and full-medial-tire labelling cuts for the recognised treads $T_1$ and $T_2$. Drawn by \texttt {experiments/draw\_medial\_tire\_cut.py} with \texttt {--whole --min-degree 5}.}}{5}{}\protected@file@percent }
\newlabel{fig:whole-medial}{{3}{5}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
\gdef \@abspage@last{5}
papers/medial_tire_cuts/paper.pdf
BIN
View File
Binary file not shown.
papers/medial_tire_cuts/paper.tex
-438
View File
@@ -1,438 +0,0 @@
%% filename: amsart-template.tex
%% American Mathematical Society
%% AMS-LaTeX v.2 template for use with amsart
%% ====================================================================
\documentclass{amsart}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\begin{document}
\title{Medial Tire Cuts}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}
\date{}
\dedicatory{}
\begin{abstract}
Starting from the medial tire decomposition of a plane triangulation, we
study the cuts that medial tires make in the full medial graph. We will
show how to use medial tires to decompose the medial graph into a tree of
three faces.
\end{abstract}
\maketitle
\section{Introduction}
This paper builds on the medial tire decomposition
of~\cite{bauerfeld-medial-tire}. For a plane triangulation $G$ with
fixed embedding we use freely the terminology and notation introduced
there: the full medial graph $M(G)$, its decomposition into full medial
tire graphs $\mathsf{M}(T)$ indexed by the treads $T$ of the tire tree
$\mathcal{T}(G,S)$ at a level source $S$, the annular medial cycle
$A(T)$, and the boundary medial vertex sets.
We will show how to use medial tires to decompose the medial graph into
a tree of three faces.
\section{Cutting a full medial tire graph}
We first describe a procedure that simultaneously \emph{labels} and
\emph{cuts} a single full medial tire graph $\mathsf{M}(T)$ so that,
after the cuts, the only faces are the outer face and $3$-faces
(triangles)---the teeth of~\cite{bauerfeld-medial-tire}. The labelling
assigns to each tooth an integer \emph{walk depth}; the cuts break the
cyclic adjacencies of the teeth so that what remains is a tree of
$3$-faces.
By a \emph{cut} we mean the duplication of a single vertex of
$\mathsf{M}(T)$: the vertex is split into two copies and the embedding is
slit open along it (a planar unzip), separating the faces that meet only
at that vertex. A cut therefore reduces the number of bounded faces that
are not teeth.
Throughout we use the teeth, up and down teeth, apexes, bites, the
annular medial cycle $A(T)$, and the auxiliary plane graph $B(T)$
of~\cite{bauerfeld-medial-tire}. Each tooth is a $3$-face of
$\mathsf{M}(T)$, and the inner faces of $B(T)$ (the root face and the
bite inner-gap faces) are the larger faces to be cut into teeth.
\begin{definition}[Walk-depth labelling and cut]
\label{def:walk-depth-cut}
Let $\mathsf{M}(T)$ be a full medial tire graph. Assign walk depths and
cuts as follows.
\begin{enumerate}
\item Pick an arbitrary up tooth, the \emph{entry tooth}. It has walk
depth $d$.
\item Traverse all the teeth that bound the inner face incident to the
entry tooth clockwise until we reach the entry tooth, incrementing the
walk depth by $1$ for each tooth traversed. (The \emph{inner face
incident to the entry tooth} is the inner face of $B(T)$ whose boundary
contains the annular edge of $A(T)$ carrying the entry tooth.)
\item When you reach the last tooth in the face, perform a \emph{cut}
by duplicating the annular vertex at which the traversal closes---the
annular vertex of $A(T)$ shared by the last tooth and the entry tooth.
\item Find the tooth $t$ with the highest walk depth which is a member
of a bite.
\item If $t$ is incident to a face $F$ with unlabelled teeth, traverse
the teeth in $F$ starting from $t$ in the direction of the tooth
incident to $t$ which is unlabelled, and increment the walk depth by
$1$ as you travel. (Here a tooth is \emph{incident to $t$} when it
shares an annular vertex of $A(T)$ with $t$.)
\item Repeat steps (3)--(5) until all teeth have been labelled.
\end{enumerate}
\end{definition}
\begin{remark}[Closing tooth of a descended face]
\label{rem:closing-tooth}
For the entry face the traversal of step (2) closes by returning to the
entry tooth, so the cut of step (3) duplicates the annular vertex shared
by the last tooth and the entry tooth. For a face $F$ entered in step
(5), the traversal instead closes upon reaching an already-labelled
tooth: the other tooth of the bite through which $F$ was entered. In
both cases the cut of step (3) duplicates the annular vertex shared by
the last newly labelled tooth and this \emph{closing tooth}. Since both
teeth of a bite are labelled while traversing its parent face, every
descended face closes on such a tooth.
\end{remark}
\begin{example}[A worked walk-depth labelling and cut]
\label{ex:worked-cut}
Figure~\ref{fig:worked-cut} shows a full medial tire graph with annular
cycle of length $8$, generated by the full medial tire generator
of~\cite{bauerfeld-medial-tire}. Its eight teeth are: three up teeth on
the annular edges $5,6,7$ in the root face; one bite pairing the annular
edges $0$ and $4$; and three singleton down teeth on the annular edges
$1,2,3$ lying in that bite's inner-gap face.
Take the up tooth on edge $5$ as the entry tooth, with walk depth $0$.
Its inner face is the root face, bounded by the teeth on edges
$5,6,7,0,4$ in clockwise order. Step (2) labels them
\[
5\mapsto 0,\quad 6\mapsto 1,\quad 7\mapsto 2,\quad
0\mapsto 3,\quad 4\mapsto 4,
\]
and step (3) cuts by duplicating the annular vertex $a_5$ shared by the
last tooth (edge $4$) and the entry tooth (edge $5$). The highest-depth
bite tooth is now the one on edge $4$ (walk depth $4$); it is incident to
the still-unlabelled inner-gap face of the bite $(0,4)$. Entering that
face from edge $4$ toward its unlabelled neighbour, step (5) labels the
three down teeth
\[
3\mapsto 5,\quad 2\mapsto 6,\quad 1\mapsto 7,
\]
and closes on the already-labelled bite tooth of edge $0$, so step (3)
cuts by duplicating the annular vertex $a_1$
(Remark~\ref{rem:closing-tooth}). All eight teeth are now labelled, and
the two cuts leave only the outer face and the eight teeth as
$3$-faces. The labelling and cuts are produced by the script
\texttt{experiments/medial\_tire\_cut\_labelling.py}.
\end{example}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.6,
ann/.style={circle, fill=black, inner sep=1.0pt},
upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
cyc/.style={black, line width=1.0pt},
tth/.style={black!55, line width=0.4pt},
lbl/.style={font=\scriptsize},
dlbl/.style={font=\scriptsize\bfseries, text=black},
cut/.style={red!80!black, line width=1.3pt},
cutlbl/.style={font=\tiny, text=red!75!black}]
\draw[cyc] (0.000,1.000)--(0.707,0.707)--(1.000,0.000)--(0.707,-0.707)--(0.000,-1.000)--(-0.707,-0.707)--(-1.000,-0.000)--(-0.707,0.707)--cycle;
\draw[tth] (-1.349,-0.559)--(-0.707,-0.707) (-1.349,-0.559)--(-1.000,-0.000);
\draw[tth] (-1.349,0.559)--(-1.000,-0.000) (-1.349,0.559)--(-0.707,0.707);
\draw[tth] (-0.559,1.349)--(-0.707,0.707) (-0.559,1.349)--(0.000,1.000);
\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
\draw[tth] (0.554,-0.230)--(1.000,0.000) (0.554,-0.230)--(0.707,-0.707);
\draw[tth] (0.230,-0.554)--(0.707,-0.707) (0.230,-0.554)--(0.000,-1.000);
\draw[tth] (0.000,-0.000)--(0.000,1.000) (0.000,-0.000)--(0.707,0.707);
\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
\node[ann] at (0.000,1.000) {};
\node[ann] at (0.707,0.707) {};
\node[ann] at (1.000,0.000) {};
\node[ann] at (0.707,-0.707) {};
\node[ann] at (0.000,-1.000) {};
\node[ann] at (-0.707,-0.707) {};
\node[ann] at (-1.000,-0.000) {};
\node[ann] at (-0.707,0.707) {};
\node[upv] at (-1.349,-0.559) {};
\node[upv] at (-1.349,0.559) {};
\node[upv] at (-0.559,1.349) {};
\node[downv] at (0.554,0.230) {};
\node[downv] at (0.554,-0.230) {};
\node[downv] at (0.230,-0.554) {};
\node[bitev] at (0.000,-0.000) {};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=1.6,
ann/.style={circle, fill=black, inner sep=1.0pt},
upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
cyc/.style={black, line width=1.0pt},
tth/.style={black!55, line width=0.4pt},
lbl/.style={font=\scriptsize},
dlbl/.style={font=\scriptsize\bfseries, text=black},
cut/.style={red!80!black, line width=1.3pt},
cutlbl/.style={font=\tiny, text=red!75!black}]
\draw[cyc] (0.000,1.000)--(0.707,0.707)--(1.000,0.000)--(0.707,-0.707)--(0.000,-1.000)--(-0.707,-0.707)--(-1.000,-0.000)--(-0.707,0.707)--cycle;
\draw[tth] (-1.349,-0.559)--(-0.707,-0.707) (-1.349,-0.559)--(-1.000,-0.000);
\draw[tth] (-1.349,0.559)--(-1.000,-0.000) (-1.349,0.559)--(-0.707,0.707);
\draw[tth] (-0.559,1.349)--(-0.707,0.707) (-0.559,1.349)--(0.000,1.000);
\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
\draw[tth] (0.554,-0.230)--(1.000,0.000) (0.554,-0.230)--(0.707,-0.707);
\draw[tth] (0.230,-0.554)--(0.707,-0.707) (0.230,-0.554)--(0.000,-1.000);
\draw[tth] (0.000,-0.000)--(0.000,1.000) (0.000,-0.000)--(0.707,0.707);
\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
\node[ann] at (0.000,1.000) {};
\node[ann] at (0.707,0.707) {};
\node[ann] at (1.000,0.000) {};
\node[ann] at (0.707,-0.707) {};
\node[ann] at (0.000,-1.000) {};
\node[ann] at (-0.707,-0.707) {};
\node[ann] at (-1.000,-0.000) {};
\node[ann] at (-0.707,0.707) {};
\node[upv] at (-1.349,-0.559) {};
\node[upv] at (-1.349,0.559) {};
\node[upv] at (-0.559,1.349) {};
\node[downv] at (0.554,0.230) {};
\node[downv] at (0.554,-0.230) {};
\node[downv] at (0.230,-0.554) {};
\node[bitev] at (0.000,-0.000) {};
\node[dlbl] at (0.177,0.427) {3};
\node[dlbl] at (0.704,0.292) {7};
\node[dlbl] at (0.704,-0.292) {6};
\node[dlbl] at (0.292,-0.704) {5};
\node[dlbl] at (-0.177,-0.427) {4};
\node[dlbl] at (-1.101,-0.456) {0};
\node[dlbl] at (-1.101,0.456) {1};
\node[dlbl] at (-0.456,1.101) {2};
\draw[cut] (-0.594,-0.594)--(-0.820,-0.820);
\node[cutlbl] at (-0.919,-0.919) {cut 1};
\draw[cut] (0.594,0.594)--(0.820,0.820);
\node[cutlbl] at (0.919,0.919) {cut 2};
\node[lbl, text=blue!60!black] at (-1.663,-0.689) {entry};
\end{tikzpicture}
\caption{A full medial tire graph (left) and its walk-depth labelling and
cut (right), from Example~\ref{ex:worked-cut}. Black vertices are the
annular medial vertices of the cycle $A(T)$; blue vertices are up-tooth
apexes, red vertices are down-tooth apexes, and the larger red vertex is
the shared apex of the bite on annular edges $0$ and $4$. On the right,
each tooth carries its walk depth, and the two red slits mark the cuts:
\emph{cut~1} duplicates $a_5$ as the root-face traversal closes, and
\emph{cut~2} duplicates $a_1$ as the bite's inner-gap face closes. After
the cuts the only bounded faces are the eight teeth.}
\label{fig:worked-cut}
\end{figure}
\section{Chaining across the tire tree}
Definition~\ref{def:walk-depth-cut} labels and cuts a single full medial
tire graph. We extend it to the whole medial graph $M(G)$ through the
medial tire decomposition of~\cite{bauerfeld-medial-tire}: the tire tree
decomposes $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one per
tread $T$, glued along their boundary medial vertices. A parent tread's
inner level cycle is a child tread's outer level cycle, and the boundary
medial vertices on that shared cycle belong to both treads.
The key incidence is this. A \emph{boundary} (singleton) down tooth of a
parent tread and the up tooth of the child tread glued to it across the
shared level cycle have the \emph{same apex}: both apexes are the same
medial vertex of $M(G)$, namely the medial vertex of an edge with both
endpoints on the shared level cycle. We use this to carry the walk depth
from a parent into its children.
We label tread by tread, outward from the root:
\begin{itemize}
\item a tread with no parent in the decomposition---in particular the
innermost recognised tread---is treated as a \emph{root} and entered at
an arbitrary up tooth with walk depth $0$;
\item a child tread is entered at the up tooth whose apex is the parent's
boundary down tooth of lowest walk depth; that entry up tooth's walk depth
is one more than that down tooth's, and the walk then increments locally
within the child as in Definition~\ref{def:walk-depth-cut}.
\end{itemize}
The source cap contributes one additional cut before the recognised
treads are assembled. If the root tread enters at an up tooth whose apex
is the cap down tooth $xy$, we cut the cap annular vertex corresponding
to the counter-clockwise source edge incident to $xy$. In the example of
Figure~\ref{fig:whole-medial}, the root entry apex is the cap down tooth
$14\!-\!4$, so the cap cut is placed at the medial vertex $14\!-\!5$.
\begin{remark}[Candidate down teeth for chaining]
\label{rem:chaining-candidates}
The down teeth eligible to fix a child's entry are exactly the
\emph{boundary} (singleton) down teeth of the parent: those lying in a
single tread face, whose apex is the shared boundary medial vertex glued to
a child up tooth. A bite's two down teeth are \emph{not} eligible. By the
definition of a bite in~\cite{bauerfeld-medial-tire} its annular edge borders
two tread faces, so a bite tooth is interior to the parent tread and its
apex is a boundary medial vertex of no child. Hence ``the down tooth of
lowest walk depth'' is read among the boundary down teeth only; a bite of
even lower walk depth is skipped.
\end{remark}
Applying every tread's cuts to $M(G)$ assembles the per-tread labellings
and cuts into a single cut graph of $M(G)$ together with a global
walk-depth label map. This pipeline---random maximal planar graph, medial
graph, tire decomposition at a vertex level source, and chained walk-depth
labelling and cut---is carried out by the experiment script
\texttt{experiments/run\_medial\_tire\_cut\_experiment.py}.
\begin{example}[A medial tire cut from a random graph]
\label{ex:real-cut}
Run on a random maximal planar graph on $20$ vertices (seed $72$, level
source vertex $9$), the experiment yields a single recognised tread
$T_2$, drawn in Figure~\ref{fig:real-cut} with the walk-depth labelling
and cut emitted by the graphics companion
\texttt{experiments/draw\_medial\_tire\_cut.py}. Its annular cycle has
length $8$, with up teeth on annular edges $0,3,4$, singleton down teeth
on $1,6,7$, and a bite on the non-incident annular edges $2$ and $5$ (the
central shared apex). Entering at the up tooth on edge $0$ with walk
depth $0$, the root face is labelled in order ($0,1,2$ then $3,4,5$) and
\emph{cut~1} duplicates $a_0$ as it closes; the walk then descends through
the bite into its inner-gap face, labelling the two teeth there ($6,7$),
and \emph{cut~2} duplicates $a_3$ as that face closes. The two cuts leave
only the outer face and the eight teeth as $3$-faces.
\end{example}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.6,
ann/.style={circle, fill=black, inner sep=1.0pt},
upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
cyc/.style={black, line width=1.0pt},
tth/.style={black!55, line width=0.4pt},
lbl/.style={font=\scriptsize},
dlbl/.style={font=\scriptsize\bfseries, text=black},
cut/.style={red!80!black, line width=1.3pt},
cutlbl/.style={font=\tiny, text=red!75!black}]
\draw[cyc] (0.000,1.000)--(0.707,0.707)--(1.000,0.000)--(0.707,-0.707)--(0.000,-1.000)--(-0.707,-0.707)--(-1.000,-0.000)--(-0.707,0.707)--cycle;
\draw[tth] (0.559,1.349)--(0.000,1.000) (0.559,1.349)--(0.707,0.707);
\draw[tth] (0.559,-1.349)--(0.707,-0.707) (0.559,-1.349)--(0.000,-1.000);
\draw[tth] (-0.559,-1.349)--(0.000,-1.000) (-0.559,-1.349)--(-0.707,-0.707);
\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
\draw[tth] (-0.554,0.230)--(-1.000,-0.000) (-0.554,0.230)--(-0.707,0.707);
\draw[tth] (-0.230,0.554)--(-0.707,0.707) (-0.230,0.554)--(0.000,1.000);
\draw[tth] (0.000,-0.318)--(1.000,0.000) (0.000,-0.318)--(0.707,-0.707);
\draw[tth] (0.000,-0.318)--(-0.707,-0.707) (0.000,-0.318)--(-1.000,-0.000);
\node[ann] at (0.000,1.000) {};
\node[ann] at (0.707,0.707) {};
\node[ann] at (1.000,0.000) {};
\node[ann] at (0.707,-0.707) {};
\node[ann] at (0.000,-1.000) {};
\node[ann] at (-0.707,-0.707) {};
\node[ann] at (-1.000,-0.000) {};
\node[ann] at (-0.707,0.707) {};
\node[upv] at (0.559,1.349) {};
\node[upv] at (0.559,-1.349) {};
\node[upv] at (-0.559,-1.349) {};
\node[downv] at (0.554,0.230) {};
\node[downv] at (-0.554,0.230) {};
\node[downv] at (-0.230,0.554) {};
\node[bitev] at (0.000,-0.318) {};
\node[dlbl] at (0.456,1.101) {0};
\node[dlbl] at (0.704,0.292) {1};
\node[dlbl] at (0.427,-0.336) {2};
\node[dlbl] at (0.456,-1.101) {7};
\node[dlbl] at (-0.456,-1.101) {6};
\node[dlbl] at (-0.427,-0.336) {3};
\node[dlbl] at (-0.704,0.292) {4};
\node[dlbl] at (-0.292,0.704) {5};
\draw[cut] (0.000,0.840)--(0.000,1.160);
\node[cutlbl] at (0.000,1.300) {cut 1};
\draw[cut] (0.594,-0.594)--(0.820,-0.820);
\node[cutlbl] at (0.919,-0.919) {cut 2};
\node[lbl, text=blue!60!black] at (0.689,1.663) {entry};
\end{tikzpicture}
\caption{The recognised tread $T_2$ of the medial tire decomposition of a
random maximal planar graph on $20$ vertices
(Example~\ref{ex:real-cut}), with its walk-depth labelling and cut. Black
vertices are the annular medial vertices of $A(T)$; blue vertices are
up-tooth apexes and red vertices down-tooth apexes, the larger red vertex
being the shared apex of the bite on annular edges $2$ and $5$. Each
tooth carries its walk depth; the red slits are the two cuts.}
\label{fig:real-cut}
\end{figure}
Figure~\ref{fig:whole-medial} repeats the whole-medial-graph drawing on a
random maximal planar graph on $20$ vertices with minimum degree $5$
(plantri seed $59$, level source vertex $5$). The experiment recognises
two full medial tire treads, $T_1$ and $T_2$, and produces seven cuts:
one source-cap cut and six full-tread cuts. The
top panel shows the source triangulation with its level source
highlighted; the bottom panel draws $M(G)$ on the same straight-line
embedding by placing each medial vertex at the midpoint of its
corresponding source edge. Every medial vertex is labelled by that source
edge. Black vertices correspond to source edges joining consecutive
levels, and coloured vertices correspond to source edges within one level.
The red-highlighted vertices, walk-depth labels, and red slits are the
computed full-medial-tire labelling and cuts.
\begin{figure}[h]
\centering
\input{whole_medial_seed59_min5.tikz}
\caption{The source graph $G$ and the whole medial graph $M(G)$ of the
minimum-degree-$5$ maximal planar graph on $20$ vertices generated by
\texttt{plantri -m5} at seed $59$. The source vertex $5$ is highlighted
in the top panel. In the bottom panel, each medial vertex is placed at
the midpoint of its corresponding source edge and labelled by that edge.
Black vertices come from source edges between consecutive levels; coloured
vertices come from source edges within a single level of the chain. The
red-highlighted vertices, walk-depth labels, and seven red slits are the
computed source-cap cut and full-medial-tire labelling cuts for the
recognised treads $T_1$ and $T_2$. Drawn by
\texttt{experiments/draw\_medial\_tire\_cut.py} with
\texttt{--whole --min-degree 5}.}
\label{fig:whole-medial}
\end{figure}
\begin{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\end{thebibliography}
\end{document}
papers/medial_tire_cuts/seed59_min5_dual.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 262 KiB

papers/medial_tire_cuts/seed59_min5_dual_cut_0.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 224 KiB

papers/medial_tire_cuts/seed59_min5_dual_cut_1.png
BIN
View File
Binary file not shown.
Side by Side

Before

Width:  |  Height:  |  Size: 283 KiB

papers/medial_tire_cuts/whole_medial_seed59_min5.tikz
-403
View File
@@ -1,403 +0,0 @@
% whole medial graph: n=20 seed=59 graph_seed=59 min_degree=5 source=5 recognised treads=[1, 2] |M(G)|=54
\begin{tabular}{c}
\begin{tikzpicture}[scale=4.06,
sedge/.style={black!50, line width=0.35pt},
sv/.style={circle, draw=black!60, fill=white, inner sep=1.1pt},
srcv/.style={circle, draw=blue!75!black, fill=blue!18, line width=0.7pt, inner sep=1.8pt}]
\draw[sedge] (0.000,0.433)--(-0.500,-0.433);
\draw[sedge] (0.000,0.433)--(0.500,-0.433);
\draw[sedge] (0.000,0.433)--(0.084,-0.096);
\draw[sedge] (0.000,0.433)--(-0.018,0.026);
\draw[sedge] (0.000,0.433)--(-0.128,-0.048);
\draw[sedge] (-0.500,-0.433)--(0.500,-0.433);
\draw[sedge] (-0.500,-0.433)--(-0.128,-0.048);
\draw[sedge] (-0.500,-0.433)--(-0.073,-0.186);
\draw[sedge] (-0.500,-0.433)--(0.019,-0.303);
\draw[sedge] (0.500,-0.433)--(0.084,-0.096);
\draw[sedge] (0.500,-0.433)--(0.019,-0.303);
\draw[sedge] (0.500,-0.433)--(0.139,-0.245);
\draw[sedge] (0.084,-0.096)--(-0.018,0.026);
\draw[sedge] (0.084,-0.096)--(0.139,-0.245);
\draw[sedge] (0.084,-0.096)--(0.060,-0.177);
\draw[sedge] (0.084,-0.096)--(0.035,-0.147);
\draw[sedge] (0.084,-0.096)--(0.024,-0.131);
\draw[sedge] (0.084,-0.096)--(0.015,-0.113);
\draw[sedge] (0.084,-0.096)--(0.002,-0.076);
\draw[sedge] (-0.018,0.026)--(-0.128,-0.048);
\draw[sedge] (-0.018,0.026)--(0.002,-0.076);
\draw[sedge] (-0.018,0.026)--(-0.048,-0.081);
\draw[sedge] (-0.128,-0.048)--(-0.073,-0.186);
\draw[sedge] (-0.128,-0.048)--(-0.048,-0.081);
\draw[sedge] (-0.073,-0.186)--(0.019,-0.303);
\draw[sedge] (-0.073,-0.186)--(-0.048,-0.081);
\draw[sedge] (-0.073,-0.186)--(-0.023,-0.119);
\draw[sedge] (-0.073,-0.186)--(-0.012,-0.141);
\draw[sedge] (-0.073,-0.186)--(-0.003,-0.156);
\draw[sedge] (-0.073,-0.186)--(0.010,-0.177);
\draw[sedge] (-0.073,-0.186)--(0.031,-0.218);
\draw[sedge] (0.019,-0.303)--(0.139,-0.245);
\draw[sedge] (0.019,-0.303)--(0.031,-0.218);
\draw[sedge] (0.139,-0.245)--(0.060,-0.177);
\draw[sedge] (0.139,-0.245)--(0.031,-0.218);
\draw[sedge] (0.060,-0.177)--(0.035,-0.147);
\draw[sedge] (0.060,-0.177)--(0.010,-0.177);
\draw[sedge] (0.060,-0.177)--(0.031,-0.218);
\draw[sedge] (0.035,-0.147)--(0.024,-0.131);
\draw[sedge] (0.035,-0.147)--(-0.003,-0.156);
\draw[sedge] (0.035,-0.147)--(0.010,-0.177);
\draw[sedge] (0.024,-0.131)--(0.015,-0.113);
\draw[sedge] (0.024,-0.131)--(-0.012,-0.141);
\draw[sedge] (0.024,-0.131)--(-0.003,-0.156);
\draw[sedge] (0.015,-0.113)--(0.002,-0.076);
\draw[sedge] (0.015,-0.113)--(-0.023,-0.119);
\draw[sedge] (0.015,-0.113)--(-0.012,-0.141);
\draw[sedge] (0.002,-0.076)--(-0.048,-0.081);
\draw[sedge] (0.002,-0.076)--(-0.023,-0.119);
\draw[sedge] (-0.048,-0.081)--(-0.023,-0.119);
\draw[sedge] (-0.023,-0.119)--(-0.012,-0.141);
\draw[sedge] (-0.012,-0.141)--(-0.003,-0.156);
\draw[sedge] (-0.003,-0.156)--(0.010,-0.177);
\draw[sedge] (0.010,-0.177)--(0.031,-0.218);
\node[sv] at (0.000,0.433) {};
\node[sv] at (-0.500,-0.433) {};
\node[sv] at (0.500,-0.433) {};
\node[sv] at (0.084,-0.096) {};
\node[sv] at (-0.018,0.026) {};
\node[srcv] at (-0.128,-0.048) {};
\node[sv] at (-0.073,-0.186) {};
\node[sv] at (0.019,-0.303) {};
\node[sv] at (0.139,-0.245) {};
\node[sv] at (0.060,-0.177) {};
\node[sv] at (0.035,-0.147) {};
\node[sv] at (0.024,-0.131) {};
\node[sv] at (0.015,-0.113) {};
\node[sv] at (0.002,-0.076) {};
\node[sv] at (-0.048,-0.081) {};
\node[sv] at (-0.023,-0.119) {};
\node[sv] at (-0.012,-0.141) {};
\node[sv] at (-0.003,-0.156) {};
\node[sv] at (0.010,-0.177) {};
\node[sv] at (0.031,-0.218) {};
\node[font=\scriptsize, text=blue!70!black] at (-0.128,-0.133) {source 5};
\end{tikzpicture}
\\[-0.25ex]
{\scriptsize source graph $G$}
\\[1.0ex]
\begin{tikzpicture}[scale=7.0,
base/.style={black!12, line width=0.25pt},
med/.style={black!38, line width=0.32pt},
annv/.style={circle, draw=black!70, fill=black!18, inner sep=1.0pt},
levone/.style={circle, draw=orange!75!black, fill=orange!20, inner sep=1.2pt},
levtwo/.style={circle, draw=violet!70!black, fill=violet!18, inner sep=1.2pt},
levthree/.style={circle, draw=teal!70!black, fill=teal!18, inner sep=1.2pt},
knownv/.style={circle, draw=red!70!black, fill=red!24, inner sep=1.5pt},
elbl/.style={font=\tiny, text=black!70, inner sep=0.2pt},
dlbl/.style={font=\tiny\bfseries, text=black, inner sep=0.5pt},
cut/.style={red!80!black, line width=1.0pt},
cutlbl/.style={font=\tiny, text=red!75!black}]
\draw[base] (0.000,0.433)--(-0.500,-0.433);
\draw[base] (0.000,0.433)--(0.500,-0.433);
\draw[base] (0.000,0.433)--(0.084,-0.096);
\draw[base] (0.000,0.433)--(-0.018,0.026);
\draw[base] (0.000,0.433)--(-0.128,-0.048);
\draw[base] (-0.500,-0.433)--(0.500,-0.433);
\draw[base] (-0.500,-0.433)--(-0.128,-0.048);
\draw[base] (-0.500,-0.433)--(-0.073,-0.186);
\draw[base] (-0.500,-0.433)--(0.019,-0.303);
\draw[base] (0.500,-0.433)--(0.084,-0.096);
\draw[base] (0.500,-0.433)--(0.019,-0.303);
\draw[base] (0.500,-0.433)--(0.139,-0.245);
\draw[base] (0.084,-0.096)--(-0.018,0.026);
\draw[base] (0.084,-0.096)--(0.139,-0.245);
\draw[base] (0.084,-0.096)--(0.060,-0.177);
\draw[base] (0.084,-0.096)--(0.035,-0.147);
\draw[base] (0.084,-0.096)--(0.024,-0.131);
\draw[base] (0.084,-0.096)--(0.015,-0.113);
\draw[base] (0.084,-0.096)--(0.002,-0.076);
\draw[base] (-0.018,0.026)--(-0.128,-0.048);
\draw[base] (-0.018,0.026)--(0.002,-0.076);
\draw[base] (-0.018,0.026)--(-0.048,-0.081);
\draw[base] (-0.128,-0.048)--(-0.073,-0.186);
\draw[base] (-0.128,-0.048)--(-0.048,-0.081);
\draw[base] (-0.073,-0.186)--(0.019,-0.303);
\draw[base] (-0.073,-0.186)--(-0.048,-0.081);
\draw[base] (-0.073,-0.186)--(-0.023,-0.119);
\draw[base] (-0.073,-0.186)--(-0.012,-0.141);
\draw[base] (-0.073,-0.186)--(-0.003,-0.156);
\draw[base] (-0.073,-0.186)--(0.010,-0.177);
\draw[base] (-0.073,-0.186)--(0.031,-0.218);
\draw[base] (0.019,-0.303)--(0.139,-0.245);
\draw[base] (0.019,-0.303)--(0.031,-0.218);
\draw[base] (0.139,-0.245)--(0.060,-0.177);
\draw[base] (0.139,-0.245)--(0.031,-0.218);
\draw[base] (0.060,-0.177)--(0.035,-0.147);
\draw[base] (0.060,-0.177)--(0.010,-0.177);
\draw[base] (0.060,-0.177)--(0.031,-0.218);
\draw[base] (0.035,-0.147)--(0.024,-0.131);
\draw[base] (0.035,-0.147)--(-0.003,-0.156);
\draw[base] (0.035,-0.147)--(0.010,-0.177);
\draw[base] (0.024,-0.131)--(0.015,-0.113);
\draw[base] (0.024,-0.131)--(-0.012,-0.141);
\draw[base] (0.024,-0.131)--(-0.003,-0.156);
\draw[base] (0.015,-0.113)--(0.002,-0.076);
\draw[base] (0.015,-0.113)--(-0.023,-0.119);
\draw[base] (0.015,-0.113)--(-0.012,-0.141);
\draw[base] (0.002,-0.076)--(-0.048,-0.081);
\draw[base] (0.002,-0.076)--(-0.023,-0.119);
\draw[base] (-0.048,-0.081)--(-0.023,-0.119);
\draw[base] (-0.023,-0.119)--(-0.012,-0.141);
\draw[base] (-0.012,-0.141)--(-0.003,-0.156);
\draw[base] (-0.003,-0.156)--(0.010,-0.177);
\draw[base] (0.010,-0.177)--(0.031,-0.218);
\draw[med] (-0.250,0.000)--(-0.064,0.192);
\draw[med] (-0.250,0.000)--(0.250,0.000);
\draw[med] (-0.250,0.000)--(0.000,-0.433);
\draw[med] (-0.250,0.000)--(-0.314,-0.241);
\draw[med] (0.250,0.000)--(0.042,0.169);
\draw[med] (0.250,0.000)--(0.000,-0.433);
\draw[med] (0.250,0.000)--(0.292,-0.264);
\draw[med] (0.042,0.169)--(-0.009,0.230);
\draw[med] (0.042,0.169)--(0.292,-0.264);
\draw[med] (0.042,0.169)--(0.033,-0.035);
\draw[med] (-0.009,0.230)--(-0.064,0.192);
\draw[med] (-0.009,0.230)--(0.033,-0.035);
\draw[med] (-0.009,0.230)--(-0.073,-0.011);
\draw[med] (-0.064,0.192)--(-0.073,-0.011);
\draw[med] (-0.064,0.192)--(-0.314,-0.241);
\draw[med] (0.000,-0.433)--(-0.240,-0.368);
\draw[med] (0.000,-0.433)--(0.260,-0.368);
\draw[med] (-0.314,-0.241)--(-0.286,-0.310);
\draw[med] (-0.314,-0.241)--(-0.100,-0.117);
\draw[med] (-0.286,-0.310)--(-0.240,-0.368);
\draw[med] (-0.286,-0.310)--(-0.100,-0.117);
\draw[med] (-0.286,-0.310)--(-0.027,-0.245);
\draw[med] (-0.240,-0.368)--(-0.027,-0.245);
\draw[med] (-0.240,-0.368)--(0.260,-0.368);
\draw[med] (0.292,-0.264)--(0.319,-0.339);
\draw[med] (0.292,-0.264)--(0.111,-0.171);
\draw[med] (0.260,-0.368)--(0.319,-0.339);
\draw[med] (0.260,-0.368)--(0.079,-0.274);
\draw[med] (0.319,-0.339)--(0.079,-0.274);
\draw[med] (0.319,-0.339)--(0.111,-0.171);
\draw[med] (0.033,-0.035)--(0.043,-0.086);
\draw[med] (0.033,-0.035)--(-0.008,-0.025);
\draw[med] (0.111,-0.171)--(0.072,-0.136);
\draw[med] (0.111,-0.171)--(0.099,-0.211);
\draw[med] (0.072,-0.136)--(0.059,-0.122);
\draw[med] (0.072,-0.136)--(0.099,-0.211);
\draw[med] (0.072,-0.136)--(0.047,-0.162);
\draw[med] (0.059,-0.122)--(0.054,-0.113);
\draw[med] (0.059,-0.122)--(0.047,-0.162);
\draw[med] (0.059,-0.122)--(0.029,-0.139);
\draw[med] (0.054,-0.113)--(0.049,-0.104);
\draw[med] (0.054,-0.113)--(0.029,-0.139);
\draw[med] (0.054,-0.113)--(0.019,-0.122);
\draw[med] (0.049,-0.104)--(0.043,-0.086);
\draw[med] (0.049,-0.104)--(0.019,-0.122);
\draw[med] (0.049,-0.104)--(0.008,-0.095);
\draw[med] (0.043,-0.086)--(0.008,-0.095);
\draw[med] (0.043,-0.086)--(-0.008,-0.025);
\draw[med] (-0.073,-0.011)--(-0.033,-0.027);
\draw[med] (-0.073,-0.011)--(-0.088,-0.064);
\draw[med] (-0.008,-0.025)--(-0.033,-0.027);
\draw[med] (-0.008,-0.025)--(-0.023,-0.079);
\draw[med] (-0.033,-0.027)--(-0.023,-0.079);
\draw[med] (-0.033,-0.027)--(-0.088,-0.064);
\draw[med] (-0.100,-0.117)--(-0.088,-0.064);
\draw[med] (-0.100,-0.117)--(-0.060,-0.134);
\draw[med] (-0.088,-0.064)--(-0.060,-0.134);
\draw[med] (-0.027,-0.245)--(-0.021,-0.202);
\draw[med] (-0.027,-0.245)--(0.025,-0.260);
\draw[med] (-0.060,-0.134)--(-0.048,-0.153);
\draw[med] (-0.060,-0.134)--(-0.035,-0.100);
\draw[med] (-0.048,-0.153)--(-0.042,-0.164);
\draw[med] (-0.048,-0.153)--(-0.035,-0.100);
\draw[med] (-0.048,-0.153)--(-0.018,-0.130);
\draw[med] (-0.042,-0.164)--(-0.038,-0.171);
\draw[med] (-0.042,-0.164)--(-0.018,-0.130);
\draw[med] (-0.042,-0.164)--(-0.008,-0.149);
\draw[med] (-0.038,-0.171)--(-0.031,-0.182);
\draw[med] (-0.038,-0.171)--(-0.008,-0.149);
\draw[med] (-0.038,-0.171)--(0.003,-0.167);
\draw[med] (-0.031,-0.182)--(-0.021,-0.202);
\draw[med] (-0.031,-0.182)--(0.003,-0.167);
\draw[med] (-0.031,-0.182)--(0.021,-0.197);
\draw[med] (-0.021,-0.202)--(0.021,-0.197);
\draw[med] (-0.021,-0.202)--(0.025,-0.260);
\draw[med] (0.079,-0.274)--(0.025,-0.260);
\draw[med] (0.079,-0.274)--(0.085,-0.231);
\draw[med] (0.025,-0.260)--(0.085,-0.231);
\draw[med] (0.099,-0.211)--(0.085,-0.231);
\draw[med] (0.099,-0.211)--(0.045,-0.197);
\draw[med] (0.085,-0.231)--(0.045,-0.197);
\draw[med] (0.047,-0.162)--(0.035,-0.177);
\draw[med] (0.047,-0.162)--(0.022,-0.162);
\draw[med] (0.035,-0.177)--(0.045,-0.197);
\draw[med] (0.035,-0.177)--(0.021,-0.197);
\draw[med] (0.035,-0.177)--(0.022,-0.162);
\draw[med] (0.045,-0.197)--(0.021,-0.197);
\draw[med] (0.029,-0.139)--(0.016,-0.152);
\draw[med] (0.029,-0.139)--(0.010,-0.144);
\draw[med] (0.016,-0.152)--(0.022,-0.162);
\draw[med] (0.016,-0.152)--(0.003,-0.167);
\draw[med] (0.016,-0.152)--(0.010,-0.144);
\draw[med] (0.022,-0.162)--(0.003,-0.167);
\draw[med] (0.019,-0.122)--(0.006,-0.136);
\draw[med] (0.019,-0.122)--(0.001,-0.127);
\draw[med] (0.006,-0.136)--(0.010,-0.144);
\draw[med] (0.006,-0.136)--(-0.008,-0.149);
\draw[med] (0.006,-0.136)--(0.001,-0.127);
\draw[med] (0.010,-0.144)--(-0.008,-0.149);
\draw[med] (0.008,-0.095)--(-0.004,-0.116);
\draw[med] (0.008,-0.095)--(-0.011,-0.098);
\draw[med] (-0.004,-0.116)--(0.001,-0.127);
\draw[med] (-0.004,-0.116)--(-0.018,-0.130);
\draw[med] (-0.004,-0.116)--(-0.011,-0.098);
\draw[med] (0.001,-0.127)--(-0.018,-0.130);
\draw[med] (-0.023,-0.079)--(-0.011,-0.098);
\draw[med] (-0.023,-0.079)--(-0.035,-0.100);
\draw[med] (-0.011,-0.098)--(-0.035,-0.100);
\node[knownv] at (-0.250,0.000) {};
\node[annv] at (0.250,0.000) {};
\node[annv] at (0.042,0.169) {};
\node[knownv] at (-0.009,0.230) {};
\node[annv] at (-0.064,0.192) {};
\node[annv] at (0.000,-0.433) {};
\node[annv] at (-0.314,-0.241) {};
\node[knownv] at (-0.286,-0.310) {};
\node[annv] at (-0.240,-0.368) {};
\node[knownv] at (0.292,-0.264) {};
\node[knownv] at (0.260,-0.368) {};
\node[annv] at (0.319,-0.339) {};
\node[annv] at (0.033,-0.035) {};
\node[annv] at (0.111,-0.171) {};
\node[annv] at (0.072,-0.136) {};
\node[annv] at (-0.073,-0.011) {};
\node[annv] at (-0.100,-0.117) {};
\node[annv] at (-0.027,-0.245) {};
\node[annv] at (0.079,-0.274) {};
\node[knownv] at (0.099,-0.211) {};
\node[annv] at (0.059,-0.122) {};
\node[knownv] at (0.047,-0.162) {};
\node[knownv] at (0.029,-0.139) {};
\node[annv] at (0.016,-0.152) {};
\node[annv] at (0.022,-0.162) {};
\node[annv] at (0.054,-0.113) {};
\node[knownv] at (0.019,-0.122) {};
\node[annv] at (0.006,-0.136) {};
\node[annv] at (0.010,-0.144) {};
\node[annv] at (0.049,-0.104) {};
\node[annv] at (0.008,-0.095) {};
\node[annv] at (-0.004,-0.116) {};
\node[annv] at (0.001,-0.127) {};
\node[knownv] at (0.043,-0.086) {};
\node[annv] at (-0.008,-0.025) {};
\node[annv] at (-0.023,-0.079) {};
\node[knownv] at (-0.011,-0.098) {};
\node[knownv] at (-0.033,-0.027) {};
\node[annv] at (-0.088,-0.064) {};
\node[knownv] at (-0.060,-0.134) {};
\node[annv] at (-0.035,-0.100) {};
\node[annv] at (-0.048,-0.153) {};
\node[knownv] at (-0.018,-0.130) {};
\node[annv] at (-0.042,-0.164) {};
\node[knownv] at (-0.008,-0.149) {};
\node[annv] at (-0.038,-0.171) {};
\node[knownv] at (0.003,-0.167) {};
\node[annv] at (-0.031,-0.182) {};
\node[annv] at (0.035,-0.177) {};
\node[knownv] at (0.021,-0.197) {};
\node[annv] at (-0.021,-0.202) {};
\node[knownv] at (0.025,-0.260) {};
\node[annv] at (0.085,-0.231) {};
\node[annv] at (0.045,-0.197) {};
\node[elbl] at (-0.250,0.000) [yshift=-4.8pt] {$0\!{-}\!1$};
\node[elbl] at (0.250,0.000) [yshift=-4.8pt] {$0\!{-}\!2$};
\node[elbl] at (0.042,0.169) [yshift=-4.8pt] {$0\!{-}\!3$};
\node[elbl] at (-0.009,0.230) [yshift=-4.8pt] {$0\!{-}\!4$};
\node[elbl] at (-0.064,0.192) [yshift=-4.8pt] {$0\!{-}\!5$};
\node[elbl] at (0.000,-0.433) [yshift=-4.8pt] {$1\!{-}\!2$};
\node[elbl] at (-0.314,-0.241) [yshift=-4.8pt] {$1\!{-}\!5$};
\node[elbl] at (-0.286,-0.310) [yshift=-4.8pt] {$1\!{-}\!6$};
\node[elbl] at (-0.240,-0.368) [yshift=-4.8pt] {$1\!{-}\!7$};
\node[elbl] at (0.292,-0.264) [yshift=-4.8pt] {$2\!{-}\!3$};
\node[elbl] at (0.260,-0.368) [yshift=-4.8pt] {$2\!{-}\!7$};
\node[elbl] at (0.319,-0.339) [yshift=-4.8pt] {$2\!{-}\!8$};
\node[elbl] at (0.033,-0.035) [yshift=-4.8pt] {$3\!{-}\!4$};
\node[elbl] at (0.111,-0.171) [yshift=-4.8pt] {$3\!{-}\!8$};
\node[elbl] at (0.072,-0.136) [yshift=-4.8pt] {$3\!{-}\!9$};
\node[elbl] at (-0.073,-0.011) [yshift=-4.8pt] {$4\!{-}\!5$};
\node[elbl] at (-0.100,-0.117) [yshift=-4.8pt] {$5\!{-}\!6$};
\node[elbl] at (-0.027,-0.245) [yshift=-4.8pt] {$6\!{-}\!7$};
\node[elbl] at (0.079,-0.274) [yshift=-4.8pt] {$7\!{-}\!8$};
\node[elbl] at (0.099,-0.211) [yshift=-4.8pt] {$8\!{-}\!9$};
\node[elbl] at (0.059,-0.122) [yshift=-4.8pt] {$10\!{-}\!3$};
\node[elbl] at (0.047,-0.162) [yshift=-4.8pt] {$10\!{-}\!9$};
\node[elbl] at (0.029,-0.139) [yshift=-4.8pt] {$10\!{-}\!11$};
\node[elbl] at (0.016,-0.152) [yshift=-4.8pt] {$10\!{-}\!17$};
\node[elbl] at (0.022,-0.162) [yshift=-4.8pt] {$10\!{-}\!18$};
\node[elbl] at (0.054,-0.113) [yshift=-4.8pt] {$11\!{-}\!3$};
\node[elbl] at (0.019,-0.122) [yshift=-4.8pt] {$11\!{-}\!12$};
\node[elbl] at (0.006,-0.136) [yshift=-4.8pt] {$11\!{-}\!16$};
\node[elbl] at (0.010,-0.144) [yshift=-4.8pt] {$11\!{-}\!17$};
\node[elbl] at (0.049,-0.104) [yshift=-4.8pt] {$12\!{-}\!3$};
\node[elbl] at (0.008,-0.095) [yshift=-4.8pt] {$12\!{-}\!13$};
\node[elbl] at (-0.004,-0.116) [yshift=-4.8pt] {$12\!{-}\!15$};
\node[elbl] at (0.001,-0.127) [yshift=-4.8pt] {$12\!{-}\!16$};
\node[elbl] at (0.043,-0.086) [yshift=-4.8pt] {$13\!{-}\!3$};
\node[elbl] at (-0.008,-0.025) [yshift=-4.8pt] {$13\!{-}\!4$};
\node[elbl] at (-0.023,-0.079) [yshift=-4.8pt] {$13\!{-}\!14$};
\node[elbl] at (-0.011,-0.098) [yshift=-4.8pt] {$13\!{-}\!15$};
\node[elbl] at (-0.033,-0.027) [yshift=-4.8pt] {$14\!{-}\!4$};
\node[elbl] at (-0.088,-0.064) [yshift=-4.8pt] {$14\!{-}\!5$};
\node[elbl] at (-0.060,-0.134) [yshift=-4.8pt] {$14\!{-}\!6$};
\node[elbl] at (-0.035,-0.100) [yshift=-4.8pt] {$14\!{-}\!15$};
\node[elbl] at (-0.048,-0.153) [yshift=-4.8pt] {$15\!{-}\!6$};
\node[elbl] at (-0.018,-0.130) [yshift=-4.8pt] {$15\!{-}\!16$};
\node[elbl] at (-0.042,-0.164) [yshift=-4.8pt] {$16\!{-}\!6$};
\node[elbl] at (-0.008,-0.149) [yshift=-4.8pt] {$16\!{-}\!17$};
\node[elbl] at (-0.038,-0.171) [yshift=-4.8pt] {$17\!{-}\!6$};
\node[elbl] at (0.003,-0.167) [yshift=-4.8pt] {$17\!{-}\!18$};
\node[elbl] at (-0.031,-0.182) [yshift=-4.8pt] {$18\!{-}\!6$};
\node[elbl] at (0.035,-0.177) [yshift=-4.8pt] {$18\!{-}\!9$};
\node[elbl] at (0.021,-0.197) [yshift=-4.8pt] {$18\!{-}\!19$};
\node[elbl] at (-0.021,-0.202) [yshift=-4.8pt] {$19\!{-}\!6$};
\node[elbl] at (0.025,-0.260) [yshift=-4.8pt] {$19\!{-}\!7$};
\node[elbl] at (0.085,-0.231) [yshift=-4.8pt] {$19\!{-}\!8$};
\node[elbl] at (0.045,-0.197) [yshift=-4.8pt] {$19\!{-}\!9$};
\node[dlbl] at (-0.250,0.000) [yshift=5.0pt] {4};
\node[dlbl] at (-0.009,0.230) [yshift=5.0pt] {2};
\node[dlbl] at (-0.286,-0.310) [yshift=5.0pt] {6};
\node[dlbl] at (0.292,-0.264) [yshift=5.0pt] {3,18};
\node[dlbl] at (0.260,-0.368) [yshift=5.0pt] {5,17};
\node[dlbl] at (0.099,-0.211) [yshift=5.0pt] {13,14};
\node[dlbl] at (0.047,-0.162) [yshift=5.0pt] {11,12};
\node[dlbl] at (0.029,-0.139) [yshift=5.0pt] {7,8};
\node[dlbl] at (0.019,-0.122) [yshift=5.0pt] {5,6};
\node[dlbl] at (0.043,-0.086) [yshift=5.0pt] {1,2};
\node[dlbl] at (-0.011,-0.098) [yshift=5.0pt] {3,13};
\node[dlbl] at (-0.033,-0.027) [yshift=5.0pt] {0};
\node[dlbl] at (-0.060,-0.134) [yshift=5.0pt] {12};
\node[dlbl] at (-0.018,-0.130) [yshift=5.0pt] {4,11};
\node[dlbl] at (-0.008,-0.149) [yshift=5.0pt] {9,10};
\node[dlbl] at (0.003,-0.167) [yshift=5.0pt] {9,10};
\node[dlbl] at (0.021,-0.197) [yshift=5.0pt] {8,15};
\node[dlbl] at (0.025,-0.260) [yshift=5.0pt] {7,16};
\draw[cut] (-0.075,-0.032)--(-0.101,-0.097);
\node[cutlbl] at (-0.120,-0.142) {cut 1};
\draw[cut] (-0.026,-0.044)--(-0.020,-0.114);
\node[cutlbl] at (-0.016,-0.162) {cut 2};
\draw[cut] (0.058,-0.138)--(0.041,-0.070);
\node[cutlbl] at (0.030,-0.023) {cut 3};
\draw[cut] (-0.004,-0.102)--(0.016,-0.169);
\node[cutlbl] at (0.029,-0.217) {cut 4};
\draw[cut] (0.085,-0.146)--(0.034,-0.097);
\node[cutlbl] at (-0.001,-0.063) {cut 5};
\draw[cut] (0.035,-0.212)--(0.035,-0.142);
\node[cutlbl] at (0.034,-0.093) {cut 6};
\draw[cut] (0.079,-0.183)--(0.144,-0.158);
\node[cutlbl] at (0.190,-0.142) {cut 7};
\end{tikzpicture}
\\[-0.25ex]
{\scriptsize medial graph $M(G)$ at edge midpoints}
\end{tabular}
papers/medial_tire_cuts/whole_medial_seed72.tikz
-382
View File
@@ -1,382 +0,0 @@
% whole medial graph: n=20 seed=72 source=9 recognised treads=[2] |M(G)|=54
\begin{tabular}{c}
\begin{tikzpicture}[scale=4.06,
sedge/.style={black!50, line width=0.35pt},
sv/.style={circle, draw=black!60, fill=white, inner sep=1.1pt},
srcv/.style={circle, draw=blue!75!black, fill=blue!18, line width=0.7pt, inner sep=1.8pt}]
\draw[sedge] (0.000,0.433)--(-0.500,-0.433);
\draw[sedge] (0.000,0.433)--(0.500,-0.433);
\draw[sedge] (0.000,0.433)--(0.027,-0.257);
\draw[sedge] (0.000,0.433)--(0.199,-0.203);
\draw[sedge] (0.000,0.433)--(-0.158,-0.086);
\draw[sedge] (-0.500,-0.433)--(0.500,-0.433);
\draw[sedge] (-0.500,-0.433)--(0.027,-0.257);
\draw[sedge] (-0.500,-0.433)--(0.012,-0.355);
\draw[sedge] (-0.500,-0.433)--(-0.170,-0.348);
\draw[sedge] (-0.500,-0.433)--(-0.218,-0.346);
\draw[sedge] (-0.500,-0.433)--(-0.158,-0.086);
\draw[sedge] (-0.500,-0.433)--(-0.230,-0.345);
\draw[sedge] (0.500,-0.433)--(0.027,-0.257);
\draw[sedge] (0.500,-0.433)--(0.199,-0.203);
\draw[sedge] (0.500,-0.433)--(0.012,-0.355);
\draw[sedge] (0.500,-0.433)--(0.234,-0.280);
\draw[sedge] (0.500,-0.433)--(0.292,-0.291);
\draw[sedge] (0.500,-0.433)--(0.151,-0.341);
\draw[sedge] (0.500,-0.433)--(0.254,-0.323);
\draw[sedge] (0.027,-0.257)--(0.199,-0.203);
\draw[sedge] (0.027,-0.257)--(0.012,-0.355);
\draw[sedge] (0.027,-0.257)--(0.234,-0.280);
\draw[sedge] (0.027,-0.257)--(-0.170,-0.348);
\draw[sedge] (0.027,-0.257)--(0.151,-0.341);
\draw[sedge] (0.027,-0.257)--(0.254,-0.323);
\draw[sedge] (0.027,-0.257)--(-0.218,-0.346);
\draw[sedge] (0.027,-0.257)--(0.063,-0.317);
\draw[sedge] (0.027,-0.257)--(0.146,-0.243);
\draw[sedge] (0.027,-0.257)--(-0.158,-0.086);
\draw[sedge] (0.027,-0.257)--(0.124,-0.234);
\draw[sedge] (0.027,-0.257)--(-0.230,-0.345);
\draw[sedge] (0.199,-0.203)--(0.234,-0.280);
\draw[sedge] (0.199,-0.203)--(0.292,-0.291);
\draw[sedge] (0.199,-0.203)--(0.233,-0.249);
\draw[sedge] (0.199,-0.203)--(0.221,-0.241);
\draw[sedge] (0.199,-0.203)--(0.146,-0.243);
\draw[sedge] (0.199,-0.203)--(0.218,-0.231);
\draw[sedge] (0.199,-0.203)--(0.124,-0.234);
\draw[sedge] (0.012,-0.355)--(-0.170,-0.348);
\draw[sedge] (0.012,-0.355)--(0.151,-0.341);
\draw[sedge] (0.012,-0.355)--(0.063,-0.317);
\draw[sedge] (0.234,-0.280)--(0.292,-0.291);
\draw[sedge] (0.234,-0.280)--(0.233,-0.249);
\draw[sedge] (0.234,-0.280)--(0.254,-0.323);
\draw[sedge] (0.234,-0.280)--(0.221,-0.241);
\draw[sedge] (0.234,-0.280)--(0.146,-0.243);
\draw[sedge] (0.292,-0.291)--(0.233,-0.249);
\draw[sedge] (0.233,-0.249)--(0.221,-0.241);
\draw[sedge] (0.233,-0.249)--(0.218,-0.231);
\draw[sedge] (-0.170,-0.348)--(-0.218,-0.346);
\draw[sedge] (0.151,-0.341)--(0.063,-0.317);
\draw[sedge] (-0.218,-0.346)--(-0.230,-0.345);
\draw[sedge] (0.221,-0.241)--(0.218,-0.231);
\draw[sedge] (0.146,-0.243)--(0.124,-0.234);
\node[sv] at (0.000,0.433) {};
\node[sv] at (-0.500,-0.433) {};
\node[sv] at (0.500,-0.433) {};
\node[sv] at (0.027,-0.257) {};
\node[sv] at (0.199,-0.203) {};
\node[sv] at (0.012,-0.355) {};
\node[sv] at (0.234,-0.280) {};
\node[sv] at (0.292,-0.291) {};
\node[sv] at (0.233,-0.249) {};
\node[srcv] at (-0.170,-0.348) {};
\node[sv] at (0.151,-0.341) {};
\node[sv] at (0.254,-0.323) {};
\node[sv] at (-0.218,-0.346) {};
\node[sv] at (0.221,-0.241) {};
\node[sv] at (0.063,-0.317) {};
\node[sv] at (0.146,-0.243) {};
\node[sv] at (0.218,-0.231) {};
\node[sv] at (-0.158,-0.086) {};
\node[sv] at (0.124,-0.234) {};
\node[sv] at (-0.230,-0.345) {};
\node[font=\scriptsize, text=blue!70!black] at (-0.170,-0.433) {source 9};
\end{tikzpicture}
\\[-0.25ex]
{\scriptsize source graph $G$}
\\[1.0ex]
\begin{tikzpicture}[scale=7.0,
base/.style={black!12, line width=0.25pt},
med/.style={black!38, line width=0.32pt},
annv/.style={circle, draw=black!70, fill=black!18, inner sep=1.0pt},
levone/.style={circle, draw=orange!75!black, fill=orange!20, inner sep=1.2pt},
levtwo/.style={circle, draw=violet!70!black, fill=violet!18, inner sep=1.2pt},
levthree/.style={circle, draw=teal!70!black, fill=teal!18, inner sep=1.2pt},
knownv/.style={circle, draw=red!70!black, fill=red!24, inner sep=1.5pt},
elbl/.style={font=\tiny, text=black!70, inner sep=0.2pt},
dlbl/.style={font=\tiny\bfseries, text=black, inner sep=0.5pt},
cut/.style={red!80!black, line width=1.0pt},
cutlbl/.style={font=\tiny, text=red!75!black}]
\draw[base] (0.000,0.433)--(-0.500,-0.433);
\draw[base] (0.000,0.433)--(0.500,-0.433);
\draw[base] (0.000,0.433)--(0.027,-0.257);
\draw[base] (0.000,0.433)--(0.199,-0.203);
\draw[base] (0.000,0.433)--(-0.158,-0.086);
\draw[base] (-0.500,-0.433)--(0.500,-0.433);
\draw[base] (-0.500,-0.433)--(0.027,-0.257);
\draw[base] (-0.500,-0.433)--(0.012,-0.355);
\draw[base] (-0.500,-0.433)--(-0.170,-0.348);
\draw[base] (-0.500,-0.433)--(-0.218,-0.346);
\draw[base] (-0.500,-0.433)--(-0.158,-0.086);
\draw[base] (-0.500,-0.433)--(-0.230,-0.345);
\draw[base] (0.500,-0.433)--(0.027,-0.257);
\draw[base] (0.500,-0.433)--(0.199,-0.203);
\draw[base] (0.500,-0.433)--(0.012,-0.355);
\draw[base] (0.500,-0.433)--(0.234,-0.280);
\draw[base] (0.500,-0.433)--(0.292,-0.291);
\draw[base] (0.500,-0.433)--(0.151,-0.341);
\draw[base] (0.500,-0.433)--(0.254,-0.323);
\draw[base] (0.027,-0.257)--(0.199,-0.203);
\draw[base] (0.027,-0.257)--(0.012,-0.355);
\draw[base] (0.027,-0.257)--(0.234,-0.280);
\draw[base] (0.027,-0.257)--(-0.170,-0.348);
\draw[base] (0.027,-0.257)--(0.151,-0.341);
\draw[base] (0.027,-0.257)--(0.254,-0.323);
\draw[base] (0.027,-0.257)--(-0.218,-0.346);
\draw[base] (0.027,-0.257)--(0.063,-0.317);
\draw[base] (0.027,-0.257)--(0.146,-0.243);
\draw[base] (0.027,-0.257)--(-0.158,-0.086);
\draw[base] (0.027,-0.257)--(0.124,-0.234);
\draw[base] (0.027,-0.257)--(-0.230,-0.345);
\draw[base] (0.199,-0.203)--(0.234,-0.280);
\draw[base] (0.199,-0.203)--(0.292,-0.291);
\draw[base] (0.199,-0.203)--(0.233,-0.249);
\draw[base] (0.199,-0.203)--(0.221,-0.241);
\draw[base] (0.199,-0.203)--(0.146,-0.243);
\draw[base] (0.199,-0.203)--(0.218,-0.231);
\draw[base] (0.199,-0.203)--(0.124,-0.234);
\draw[base] (0.012,-0.355)--(-0.170,-0.348);
\draw[base] (0.012,-0.355)--(0.151,-0.341);
\draw[base] (0.012,-0.355)--(0.063,-0.317);
\draw[base] (0.234,-0.280)--(0.292,-0.291);
\draw[base] (0.234,-0.280)--(0.233,-0.249);
\draw[base] (0.234,-0.280)--(0.254,-0.323);
\draw[base] (0.234,-0.280)--(0.221,-0.241);
\draw[base] (0.234,-0.280)--(0.146,-0.243);
\draw[base] (0.292,-0.291)--(0.233,-0.249);
\draw[base] (0.233,-0.249)--(0.221,-0.241);
\draw[base] (0.233,-0.249)--(0.218,-0.231);
\draw[base] (-0.170,-0.348)--(-0.218,-0.346);
\draw[base] (0.151,-0.341)--(0.063,-0.317);
\draw[base] (-0.218,-0.346)--(-0.230,-0.345);
\draw[base] (0.221,-0.241)--(0.218,-0.231);
\draw[base] (0.146,-0.243)--(0.124,-0.234);
\draw[med] (-0.250,0.000)--(-0.079,0.174);
\draw[med] (-0.250,0.000)--(0.250,0.000);
\draw[med] (-0.250,0.000)--(0.000,-0.433);
\draw[med] (-0.250,0.000)--(-0.329,-0.259);
\draw[med] (0.250,0.000)--(0.100,0.115);
\draw[med] (0.250,0.000)--(0.000,-0.433);
\draw[med] (0.250,0.000)--(0.350,-0.318);
\draw[med] (0.014,0.088)--(-0.079,0.174);
\draw[med] (0.014,0.088)--(0.100,0.115);
\draw[med] (0.014,0.088)--(0.113,-0.230);
\draw[med] (0.014,0.088)--(-0.065,-0.171);
\draw[med] (0.100,0.115)--(0.350,-0.318);
\draw[med] (0.100,0.115)--(0.113,-0.230);
\draw[med] (-0.079,0.174)--(-0.065,-0.171);
\draw[med] (-0.079,0.174)--(-0.329,-0.259);
\draw[med] (0.000,-0.433)--(-0.244,-0.394);
\draw[med] (0.000,-0.433)--(0.256,-0.394);
\draw[med] (-0.236,-0.345)--(-0.365,-0.389);
\draw[med] (-0.236,-0.345)--(-0.329,-0.259);
\draw[med] (-0.236,-0.345)--(-0.065,-0.171);
\draw[med] (-0.236,-0.345)--(-0.102,-0.301);
\draw[med] (-0.244,-0.394)--(-0.335,-0.390);
\draw[med] (-0.244,-0.394)--(-0.079,-0.351);
\draw[med] (-0.244,-0.394)--(0.256,-0.394);
\draw[med] (-0.335,-0.390)--(-0.359,-0.389);
\draw[med] (-0.335,-0.390)--(-0.194,-0.347);
\draw[med] (-0.335,-0.390)--(-0.079,-0.351);
\draw[med] (-0.359,-0.389)--(-0.365,-0.389);
\draw[med] (-0.359,-0.389)--(-0.224,-0.345);
\draw[med] (-0.359,-0.389)--(-0.194,-0.347);
\draw[med] (-0.329,-0.259)--(-0.065,-0.171);
\draw[med] (-0.365,-0.389)--(-0.102,-0.301);
\draw[med] (-0.365,-0.389)--(-0.224,-0.345);
\draw[med] (0.264,-0.345)--(0.377,-0.378);
\draw[med] (0.264,-0.345)--(0.325,-0.387);
\draw[med] (0.264,-0.345)--(0.140,-0.290);
\draw[med] (0.264,-0.345)--(0.089,-0.299);
\draw[med] (0.350,-0.318)--(0.396,-0.362);
\draw[med] (0.350,-0.318)--(0.245,-0.247);
\draw[med] (0.256,-0.394)--(0.325,-0.387);
\draw[med] (0.256,-0.394)--(0.081,-0.348);
\draw[med] (0.367,-0.357)--(0.396,-0.362);
\draw[med] (0.367,-0.357)--(0.377,-0.378);
\draw[med] (0.367,-0.357)--(0.244,-0.302);
\draw[med] (0.367,-0.357)--(0.263,-0.286);
\draw[med] (0.396,-0.362)--(0.263,-0.286);
\draw[med] (0.396,-0.362)--(0.245,-0.247);
\draw[med] (0.325,-0.387)--(0.081,-0.348);
\draw[med] (0.325,-0.387)--(0.089,-0.299);
\draw[med] (0.377,-0.378)--(0.140,-0.290);
\draw[med] (0.377,-0.378)--(0.244,-0.302);
\draw[med] (0.113,-0.230)--(0.076,-0.246);
\draw[med] (0.113,-0.230)--(0.162,-0.218);
\draw[med] (0.019,-0.306)--(-0.071,-0.302);
\draw[med] (0.019,-0.306)--(0.045,-0.287);
\draw[med] (0.019,-0.306)--(-0.079,-0.351);
\draw[med] (0.019,-0.306)--(0.038,-0.336);
\draw[med] (0.131,-0.268)--(0.140,-0.290);
\draw[med] (0.131,-0.268)--(0.087,-0.250);
\draw[med] (0.131,-0.268)--(0.190,-0.262);
\draw[med] (0.131,-0.268)--(0.244,-0.302);
\draw[med] (-0.071,-0.302)--(-0.096,-0.301);
\draw[med] (-0.071,-0.302)--(-0.079,-0.351);
\draw[med] (-0.071,-0.302)--(-0.194,-0.347);
\draw[med] (0.089,-0.299)--(0.045,-0.287);
\draw[med] (0.089,-0.299)--(0.107,-0.329);
\draw[med] (0.140,-0.290)--(0.244,-0.302);
\draw[med] (-0.096,-0.301)--(-0.102,-0.301);
\draw[med] (-0.096,-0.301)--(-0.194,-0.347);
\draw[med] (-0.096,-0.301)--(-0.224,-0.345);
\draw[med] (0.045,-0.287)--(0.107,-0.329);
\draw[med] (0.045,-0.287)--(0.038,-0.336);
\draw[med] (0.087,-0.250)--(0.076,-0.246);
\draw[med] (0.087,-0.250)--(0.135,-0.239);
\draw[med] (0.087,-0.250)--(0.190,-0.262);
\draw[med] (0.076,-0.246)--(0.162,-0.218);
\draw[med] (0.076,-0.246)--(0.135,-0.239);
\draw[med] (-0.102,-0.301)--(-0.224,-0.345);
\draw[med] (0.217,-0.241)--(0.173,-0.223);
\draw[med] (0.217,-0.241)--(0.210,-0.222);
\draw[med] (0.217,-0.241)--(0.190,-0.262);
\draw[med] (0.217,-0.241)--(0.227,-0.260);
\draw[med] (0.245,-0.247)--(0.216,-0.226);
\draw[med] (0.245,-0.247)--(0.262,-0.270);
\draw[med] (0.216,-0.226)--(0.209,-0.217);
\draw[med] (0.216,-0.226)--(0.262,-0.270);
\draw[med] (0.216,-0.226)--(0.225,-0.240);
\draw[med] (0.210,-0.222)--(0.209,-0.217);
\draw[med] (0.210,-0.222)--(0.219,-0.236);
\draw[med] (0.210,-0.222)--(0.227,-0.260);
\draw[med] (0.173,-0.223)--(0.162,-0.218);
\draw[med] (0.173,-0.223)--(0.190,-0.262);
\draw[med] (0.173,-0.223)--(0.135,-0.239);
\draw[med] (0.209,-0.217)--(0.225,-0.240);
\draw[med] (0.209,-0.217)--(0.219,-0.236);
\draw[med] (0.162,-0.218)--(0.135,-0.239);
\draw[med] (0.081,-0.348)--(0.038,-0.336);
\draw[med] (0.081,-0.348)--(0.107,-0.329);
\draw[med] (0.038,-0.336)--(0.107,-0.329);
\draw[med] (0.263,-0.286)--(0.233,-0.265);
\draw[med] (0.263,-0.286)--(0.262,-0.270);
\draw[med] (0.233,-0.265)--(0.227,-0.260);
\draw[med] (0.233,-0.265)--(0.227,-0.245);
\draw[med] (0.233,-0.265)--(0.262,-0.270);
\draw[med] (0.227,-0.260)--(0.227,-0.245);
\draw[med] (0.227,-0.245)--(0.225,-0.240);
\draw[med] (0.227,-0.245)--(0.219,-0.236);
\draw[med] (0.225,-0.240)--(0.219,-0.236);
\node[annv] at (-0.250,0.000) {};
\node[levtwo] at (0.250,0.000) {};
\node[annv] at (0.014,0.088) {};
\node[levtwo] at (0.100,0.115) {};
\node[levtwo] at (-0.079,0.174) {};
\node[annv] at (0.000,-0.433) {};
\node[levone] at (-0.236,-0.345) {};
\node[levone] at (-0.244,-0.394) {};
\node[annv] at (-0.335,-0.390) {};
\node[levone] at (-0.359,-0.389) {};
\node[annv] at (-0.329,-0.259) {};
\node[annv] at (-0.365,-0.389) {};
\node[annv] at (0.264,-0.345) {};
\node[knownv] at (0.350,-0.318) {};
\node[annv] at (0.256,-0.394) {};
\node[knownv] at (0.367,-0.357) {};
\node[annv] at (0.396,-0.362) {};
\node[annv] at (0.113,-0.230) {};
\node[levone] at (0.019,-0.306) {};
\node[annv] at (0.131,-0.268) {};
\node[annv] at (-0.071,-0.302) {};
\node[knownv] at (0.217,-0.241) {};
\node[annv] at (0.245,-0.247) {};
\node[annv] at (0.216,-0.226) {};
\node[annv] at (-0.079,-0.351) {};
\node[annv] at (0.263,-0.286) {};
\node[annv] at (0.233,-0.265) {};
\node[knownv] at (0.262,-0.270) {};
\node[levtwo] at (0.325,-0.387) {};
\node[annv] at (0.089,-0.299) {};
\node[annv] at (0.081,-0.348) {};
\node[levtwo] at (0.107,-0.329) {};
\node[levtwo] at (0.377,-0.378) {};
\node[annv] at (0.140,-0.290) {};
\node[levtwo] at (0.244,-0.302) {};
\node[levone] at (-0.096,-0.301) {};
\node[annv] at (-0.194,-0.347) {};
\node[annv] at (-0.224,-0.345) {};
\node[annv] at (0.210,-0.222) {};
\node[annv] at (0.227,-0.260) {};
\node[knownv] at (0.227,-0.245) {};
\node[knownv] at (0.219,-0.236) {};
\node[annv] at (0.045,-0.287) {};
\node[annv] at (0.038,-0.336) {};
\node[annv] at (0.087,-0.250) {};
\node[levtwo] at (0.173,-0.223) {};
\node[levtwo] at (0.190,-0.262) {};
\node[levtwo] at (0.135,-0.239) {};
\node[annv] at (0.209,-0.217) {};
\node[knownv] at (0.225,-0.240) {};
\node[annv] at (-0.065,-0.171) {};
\node[annv] at (0.076,-0.246) {};
\node[levtwo] at (0.162,-0.218) {};
\node[annv] at (-0.102,-0.301) {};
\node[elbl] at (-0.250,0.000) [yshift=-4.8pt] {$0\!{-}\!1$};
\node[elbl] at (0.250,0.000) [yshift=-4.8pt] {$0\!{-}\!2$};
\node[elbl] at (0.014,0.088) [yshift=-4.8pt] {$0\!{-}\!3$};
\node[elbl] at (0.100,0.115) [yshift=-4.8pt] {$0\!{-}\!4$};
\node[elbl] at (-0.079,0.174) [yshift=-4.8pt] {$0\!{-}\!17$};
\node[elbl] at (0.000,-0.433) [yshift=-4.8pt] {$1\!{-}\!2$};
\node[elbl] at (-0.236,-0.345) [yshift=-4.8pt] {$1\!{-}\!3$};
\node[elbl] at (-0.244,-0.394) [yshift=-4.8pt] {$1\!{-}\!5$};
\node[elbl] at (-0.335,-0.390) [yshift=-4.8pt] {$1\!{-}\!9$};
\node[elbl] at (-0.359,-0.389) [yshift=-4.8pt] {$1\!{-}\!12$};
\node[elbl] at (-0.329,-0.259) [yshift=-4.8pt] {$1\!{-}\!17$};
\node[elbl] at (-0.365,-0.389) [yshift=-4.8pt] {$1\!{-}\!19$};
\node[elbl] at (0.264,-0.345) [yshift=-4.8pt] {$2\!{-}\!3$};
\node[elbl] at (0.350,-0.318) [yshift=-4.8pt] {$2\!{-}\!4$};
\node[elbl] at (0.256,-0.394) [yshift=-4.8pt] {$2\!{-}\!5$};
\node[elbl] at (0.367,-0.357) [yshift=-4.8pt] {$2\!{-}\!6$};
\node[elbl] at (0.396,-0.362) [yshift=-4.8pt] {$2\!{-}\!7$};
\node[elbl] at (0.113,-0.230) [yshift=-4.8pt] {$3\!{-}\!4$};
\node[elbl] at (0.019,-0.306) [yshift=-4.8pt] {$3\!{-}\!5$};
\node[elbl] at (0.131,-0.268) [yshift=-4.8pt] {$3\!{-}\!6$};
\node[elbl] at (-0.071,-0.302) [yshift=-4.8pt] {$3\!{-}\!9$};
\node[elbl] at (0.217,-0.241) [yshift=-4.8pt] {$4\!{-}\!6$};
\node[elbl] at (0.245,-0.247) [yshift=-4.8pt] {$4\!{-}\!7$};
\node[elbl] at (0.216,-0.226) [yshift=-4.8pt] {$4\!{-}\!8$};
\node[elbl] at (-0.079,-0.351) [yshift=-4.8pt] {$5\!{-}\!9$};
\node[elbl] at (0.263,-0.286) [yshift=-4.8pt] {$6\!{-}\!7$};
\node[elbl] at (0.233,-0.265) [yshift=-4.8pt] {$6\!{-}\!8$};
\node[elbl] at (0.262,-0.270) [yshift=-4.8pt] {$7\!{-}\!8$};
\node[elbl] at (0.325,-0.387) [yshift=-4.8pt] {$10\!{-}\!2$};
\node[elbl] at (0.089,-0.299) [yshift=-4.8pt] {$10\!{-}\!3$};
\node[elbl] at (0.081,-0.348) [yshift=-4.8pt] {$10\!{-}\!5$};
\node[elbl] at (0.107,-0.329) [yshift=-4.8pt] {$10\!{-}\!14$};
\node[elbl] at (0.377,-0.378) [yshift=-4.8pt] {$11\!{-}\!2$};
\node[elbl] at (0.140,-0.290) [yshift=-4.8pt] {$11\!{-}\!3$};
\node[elbl] at (0.244,-0.302) [yshift=-4.8pt] {$11\!{-}\!6$};
\node[elbl] at (-0.096,-0.301) [yshift=-4.8pt] {$12\!{-}\!3$};
\node[elbl] at (-0.194,-0.347) [yshift=-4.8pt] {$12\!{-}\!9$};
\node[elbl] at (-0.224,-0.345) [yshift=-4.8pt] {$12\!{-}\!19$};
\node[elbl] at (0.210,-0.222) [yshift=-4.8pt] {$13\!{-}\!4$};
\node[elbl] at (0.227,-0.260) [yshift=-4.8pt] {$13\!{-}\!6$};
\node[elbl] at (0.227,-0.245) [yshift=-4.8pt] {$13\!{-}\!8$};
\node[elbl] at (0.219,-0.236) [yshift=-4.8pt] {$13\!{-}\!16$};
\node[elbl] at (0.045,-0.287) [yshift=-4.8pt] {$14\!{-}\!3$};
\node[elbl] at (0.038,-0.336) [yshift=-4.8pt] {$14\!{-}\!5$};
\node[elbl] at (0.087,-0.250) [yshift=-4.8pt] {$15\!{-}\!3$};
\node[elbl] at (0.173,-0.223) [yshift=-4.8pt] {$15\!{-}\!4$};
\node[elbl] at (0.190,-0.262) [yshift=-4.8pt] {$15\!{-}\!6$};
\node[elbl] at (0.135,-0.239) [yshift=-4.8pt] {$15\!{-}\!18$};
\node[elbl] at (0.209,-0.217) [yshift=-4.8pt] {$16\!{-}\!4$};
\node[elbl] at (0.225,-0.240) [yshift=-4.8pt] {$16\!{-}\!8$};
\node[elbl] at (-0.065,-0.171) [yshift=-4.8pt] {$17\!{-}\!3$};
\node[elbl] at (0.076,-0.246) [yshift=-4.8pt] {$18\!{-}\!3$};
\node[elbl] at (0.162,-0.218) [yshift=-4.8pt] {$18\!{-}\!4$};
\node[elbl] at (-0.102,-0.301) [yshift=-4.8pt] {$19\!{-}\!3$};
\node[dlbl] at (0.350,-0.318) [yshift=5.0pt] {6};
\node[dlbl] at (0.367,-0.357) [yshift=5.0pt] {7};
\node[dlbl] at (0.217,-0.241) [yshift=5.0pt] {0};
\node[dlbl] at (0.262,-0.270) [yshift=5.0pt] {2,3};
\node[dlbl] at (0.227,-0.245) [yshift=5.0pt] {1};
\node[dlbl] at (0.219,-0.236) [yshift=5.0pt] {5};
\node[dlbl] at (0.225,-0.240) [yshift=5.0pt] {4};
\draw[cut] (0.241,-0.204)--(0.180,-0.239);
\node[cutlbl] at (0.137,-0.263) {cut 1};
\draw[cut] (0.256,-0.320)--(0.269,-0.251);
\node[cutlbl] at (0.279,-0.203) {cut 2};
\end{tikzpicture}
\\[-0.25ex]
{\scriptsize medial graph $M(G)$ at edge midpoints}
\end{tabular}
papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_analysis.py
+56 -95
View File
@@ -178,17 +178,19 @@ def extract_tread(faces, levels, d):
}
def _cycle_order(sub: nx.Graph, comp):
"""Cyclic order of a 2-regular connected component ``comp`` of ``sub``;
None if it is not a single simple cycle of length >= 3."""
csub = sub.subgraph(comp)
if csub.number_of_nodes() < 3 or any(csub.degree(v) != 2 for v in csub):
def annular_cycle_order(M: nx.Graph, annular: set):
"""Cyclic order of the annular medial vertices (they induce a cycle)."""
sub = M.subgraph(annular)
if sub.number_of_nodes() == 0 or any(sub.degree(v) != 2 for v in sub):
return None
start = next(iter(comp))
if not nx.is_connected(sub):
return None
start = next(iter(annular))
order = [start]
prev, cur = None, start
prev = None
cur = start
while True:
nbrs = [w for w in csub.neighbors(cur) if w != prev]
nbrs = [w for w in sub.neighbors(cur) if w != prev]
if not nbrs:
break
nxt = nbrs[0]
@@ -196,33 +198,7 @@ def _cycle_order(sub: nx.Graph, comp):
break
order.append(nxt)
prev, cur = cur, nxt
return order if len(order) == csub.number_of_nodes() else None
def annular_cycle_order(M: nx.Graph, annular: set):
"""Cyclic order of the annular medial vertices when they induce a *single*
cycle; None otherwise. See ``annular_cycle_components`` for the
multi-component case."""
sub = M.subgraph(annular)
if not annular or not nx.is_connected(sub):
return None
return _cycle_order(sub, set(annular))
def annular_cycle_components(M: nx.Graph, annular: set):
"""Cyclic orders of the annular medial vertices, one per connected
component of the annular subgraph.
A tread's annular frontier may split into several disjoint cycles (one per
boundary component); each is its own full medial tire graph. Components
that are not a single simple cycle of length >= 3 are skipped."""
sub = M.subgraph(annular)
orders = []
for comp in nx.connected_components(sub):
order = _cycle_order(sub, comp)
if order is not None:
orders.append(order)
return orders
return order if len(order) == len(annular) else None
# --------------------------------------------------------------------------- #
@@ -250,35 +226,38 @@ def _linear_cut(n, bite_pairs):
return None
def _recognise_one(M, order, up, ann_global):
"""Recognise a single annular cycle (given as the cyclic order of its
medial vertices) as a ``FullMedialTireGraph``.
def recognise(M, tread):
"""Return (FullMedialTireGraph, bijection fmt-name -> medial vertex) or None.
``up`` is the tread's up-edge medial-vertex set; ``ann_global`` is the full
annular set of the tread (used to exclude annular vertices, including those
of *other* components, when picking each cycle edge's apex). Returns
``(g, bij)`` or None."""
n = len(order)
if n < 3:
``M`` here is the tread-face model M(T) (cycle + teeth + bites)."""
annular = tread["annular"]
order = annular_cycle_order(M, annular)
if order is None or len(order) < 3:
return None
ann_set = set(order)
n = len(order)
ann_set = set(annular)
apex_of_edge = []
for i in range(n):
a, b = order[i], order[(i + 1) % n]
common = [w for w in set(M.neighbors(a)) & set(M.neighbors(b))
if w not in ann_global]
common = [w for w in set(M.neighbors(a)) & set(M.neighbors(b)) if w not in ann_set]
if len(common) != 1:
return None
apex_of_edge.append(common[0])
up = set(tread["up"])
# bite apex: serves two cycle edges (== adjacent to four annular vertices)
apex_positions = defaultdict(list)
for i, ap in enumerate(apex_of_edge):
apex_positions[ap].append(i)
bite_pairs = [tuple(sorted(positions))
for positions in apex_positions.values() if len(positions) == 2]
tooth = ["U" if ap in up else "D" for ap in apex_of_edge]
tooth = []
bite_pairs = []
for ap, positions in apex_positions.items():
if len(positions) == 2:
bite_pairs.append(tuple(sorted(positions)))
for i, ap in enumerate(apex_of_edge):
tooth.append("U" if ap in up else "D")
cut = _linear_cut(n, bite_pairs)
if cut is None:
@@ -300,35 +279,14 @@ def _recognise_one(M, order, up, ann_global):
for (i, j) in sorted(g.bites):
bij[f"p{i}_{j}"] = apex_of_edge[(i + r) % n]
# verify the reconstructed graph is edge-faithful to this cycle's sub-model
# (its annular vertices together with their tooth apexes).
sub_nodes = ann_set | set(apex_of_edge)
sub_edges = {ekey(*e) for e in M.subgraph(sub_nodes).edges()}
# verify the reconstructed graph is edge-faithful to the tread-face M(T)
mt_edges = {ekey(*e) for e in M.edges()}
rec_edges = {ekey(bij[u], bij[v]) for u, v in g.edges()}
if rec_edges != sub_edges:
if rec_edges != mt_edges:
return None
return g, bij
def recognise(M, tread):
"""Recognise the tread's medial-tire structure.
A tread's annular frontier may be several disjoint cycles, each its own
full medial tire graph. Returns a list of ``(FullMedialTireGraph,
bijection fmt-name -> medial vertex)`` -- one per annular cycle component
that recognises -- or ``[]`` if none do.
``M`` here is the tread-face model M(T) (cycle(s) + teeth + bites)."""
up = set(tread["up"])
ann_global = set(tread["annular"])
tires = []
for order in annular_cycle_components(M, tread["annular"]):
rec = _recognise_one(M, order, up, ann_global)
if rec is not None:
tires.append(rec)
return tires
def canonical(coloring, ordered):
remap, out = {}, []
for v in ordered:
@@ -383,29 +341,32 @@ def iter_pieces(seed: int, color_limit: int = 400000):
if tread is None or len(tread["up"]) < 3:
continue
mt = medial_tire_facemodel(tread["tread_faces"])
for comp, (g, bij) in enumerate(recognise(mt, tread)):
mt_nodes = list(bij.values())
name_of = {v: k for k, v in bij.items()}
rec = recognise(mt, tread)
if rec is None:
continue
g, bij = rec
mt_nodes = list(bij.values())
name_of = {v: k for k, v in bij.items()}
realized = set()
for col in global_colorings:
realized.add(canonical({v: col[v] for v in mt_nodes}, mt_nodes))
realized = set()
for col in global_colorings:
realized.add(canonical({v: col[v] for v in mt_nodes}, mt_nodes))
colorings = []
seen = set()
for col in proper_3_colorings_subgraph(mt, mt_nodes):
key = canonical(col, mt_nodes)
if key in seen:
continue
seen.add(key)
fmt_col = {name_of[v]: c for v, c in col.items()}
balanced = kempe_classify(g, fmt_col).valid
is_real = key in realized
cat = ("Invalid" if not balanced
else "Realized" if is_real else "Unrealized")
colorings.append((fmt_col, cat))
meta = {"source": s, "tread": d, "comp": comp}
yield (meta, g, colorings)
colorings = []
seen = set()
for col in proper_3_colorings_subgraph(mt, mt_nodes):
key = canonical(col, mt_nodes)
if key in seen:
continue
seen.add(key)
fmt_col = {name_of[v]: c for v, c in col.items()}
balanced = kempe_classify(g, fmt_col).valid
is_real = key in realized
cat = ("Invalid" if not balanced
else "Realized" if is_real else "Unrealized")
colorings.append((fmt_col, cat))
meta = {"source": s, "tread": d}
yield (meta, g, colorings)
def analyse(seed: int, color_limit: int = 400000):
papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
+10 -17
View File
@@ -1,4 +1,5 @@
\relax
\citation{bauerfeld-medial-pigeonhole}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-nested-tire-decompositions}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
@@ -13,7 +14,7 @@
\newlabel{thm:annular-medial-colour-bound}{{3.3}{3}}
\newlabel{def:annular-teeth}{{3.4}{3}}
\newlabel{rem:teeth-sharing}{{3.5}{3}}
\newlabel{rem:up-teeth-count}{{3.6}{3}}
\newlabel{rem:up-teeth-count}{{3.6}{4}}
\newlabel{def:bite}{{3.7}{4}}
\newlabel{rem:bite-face-count}{{3.8}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A full medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{4}{}\protected@file@percent }
@@ -26,25 +27,17 @@
\newlabel{fig:medial-annular-cycle-counterexample}{{3}{5}}
\newlabel{def:medial-restriction-relation}{{3.10}{5}}
\citation{bauerfeld-nested-tire-decompositions}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{6}{}\protected@file@percent }
\newlabel{cor:medial-tire-decomposition}{{4.1}{6}}
\newlabel{def:compatible-family}{{4.2}{6}}
\newlabel{prop:gluing-criterion}{{4.3}{6}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{6}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{5.1}{6}}
\newlabel{conj:medial-chain-pigeonhole}{{5.2}{7}}
\newlabel{conj:medial-route-fct}{{5.3}{7}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{7}{}\protected@file@percent }
\newlabel{lem:kempe-cycles}{{5.5}{7}}
\newlabel{lem:kempe-conservation}{{5.6}{8}}
\newlabel{def:kempe-balanced}{{5.7}{8}}
\newlabel{rem:kempe-balance-necessary}{{5.8}{8}}
\bibcite{bauerfeld-nested-tire-decompositions}{1}
\bibcite{tait-original}{2}
\bibcite{bauerfeld-medial-pigeonhole}{2}
\bibcite{tait-original}{3}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{29.38873pt}
\newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent }
\gdef \@abspage@last{11}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{6}{}\protected@file@percent }
\newlabel{cor:medial-tire-decomposition}{{4.1}{6}}
\newlabel{def:compatible-family}{{4.2}{6}}
\newlabel{prop:gluing-criterion}{{4.3}{6}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent }
\gdef \@abspage@last{6}
papers/medial_tire_decompositions_of_plane_triangulations/paper.log
+30 -30
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 11 JUN 2026 16:44
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 JUN 2026 21:07
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -496,7 +496,7 @@ e
))
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
[2] [3]
Overfull \hbox (62.13657pt too wide) in paragraph at lines 363--372
Overfull \hbox (62.13657pt too wide) in paragraph at lines 366--375
[][]
[]
@@ -506,39 +506,39 @@ LaTeX Warning: `h' float specifier changed to `ht'.
LaTeX Warning: `h' float specifier changed to `ht'.
[4] [5] [6] [7] [8] [9] [10] [11] (./paper.aux) )
[4] [5] [6] (./paper.aux) )
Here is how much of TeX's memory you used:
14419 strings out of 478268
283755 string characters out of 5846347
609349 words of memory out of 5000000
32248 multiletter control sequences out of 15000+600000
14413 strings out of 478268
283613 string characters out of 5846347
608323 words of memory out of 5000000
32242 multiletter control sequences out of 15000+600000
477048 words of font info for 58 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191
84i,8n,89p,736b,838s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf
-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-
dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-di
st/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-di
st/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dis
t/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/
fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fo
nts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/font
s/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/
type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
e1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
public/amsfonts/cm/cmss10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
blic/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
ic/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
msfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
onts/cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/symbols/msam10.pfb>
Output written on paper.pdf (11 pages, 277538 bytes).
</usr/local/texlive/2022/texmf-dist/fonts/type1/publ
ic/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
msfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
fonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
nts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
s/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/
cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cms
s10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy
10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5
.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.p
fb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb>
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></
usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb></us
r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (6 pages, 250239 bytes).
PDF statistics:
138 PDF objects out of 1000 (max. 8388607)
86 compressed objects within 1 object stream
121 PDF objects out of 1000 (max. 8388607)
74 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
13 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf
BIN
View File
Binary file not shown.
papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
+11 -337
View File
@@ -53,9 +53,12 @@ isomorphic to the medial graph of the planar dual $G^*$, and proper
$3$-vertex-colourings of $M(G)$ are equivalent to proper
$3$-edge-colourings of the cubic dual. Thus Tait's reformulation of
the Four Colour Theorem may be studied through proper vertex
$3$-colourings of medial subgraphs. We define medial tire pieces,
their boundary-state restriction relations, and a chain-pigeonhole
conjecture for compatible medial boundary states across the tire tree.
$3$-colourings of medial subgraphs. We define medial tire pieces and
their boundary-state restriction relations, and show that a proper
vertex $3$-colouring of $M(G)$ amounts to a compatible selection of
these boundary states across the tire tree. The resulting
pigeonhole programme for the Four Colour Theorem is developed
in~\cite{bauerfeld-medial-pigeonhole}.
\end{abstract}
\maketitle
@@ -656,340 +659,6 @@ properness is already enforced by one of the local colourings. Hence
$\varphi$ is a proper vertex $3$-colouring of $M(G)$.
\end{proof}
\section{A medial pigeonhole programme}
The restriction relation $R_T$ records exactly the local information
needed to pass a medial $3$-colouring through a tire. In a nested
chain
\[
T_0 \supset T_1 \supset \cdots \supset T_k,
\]
the outer boundary state of $T_{i+1}$ must match an inner boundary
state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.
\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
subgraph induced by that boundary set, considered up to permutation of
the three colours and the dihedral symmetries of the boundary walk
when that boundary is a cycle.
\end{definition}
\begin{conjecture}[Medial chain-pigeonhole principle]
\label{conj:medial-chain-pigeonhole}
There is a function $N(k)$ such that the following holds. Let
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
tire treads whose relevant boundary medial walks have length at most
$k$. Then two adjacent restriction relations in the chain have
compatible medial boundary states after colour permutation and boundary
symmetry. Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint proper vertex $3$-colouring
restrictions.
\end{conjecture}
\begin{conjecture}[Medial tire route to the Four Colour Theorem]
\label{conj:medial-route-fct}
For every plane triangulation $G$ and every level source $S$, the
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
compatible selection of boundary states across the tire tree. Hence
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
\end{conjecture}
\begin{remark}
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
the Four Colour Theorem when combined with Tait's correspondence. The
point of the formulation is not to weaken the target theorem, but to
move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}
\subsection{Kempe-cycle conservation across medial tires}
We now record an additional structure carried by proper
$3$-colourings of medial graphs. This structure will be useful for
describing how colourings glue across level cycles.
Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
\varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$. For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.
Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
cycle. We call these components the $P$-Kempe cycles of $\varphi$.
\begin{lemma}[Kempe chains are cycles]
\label{lem:kempe-cycles}
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$. For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}
\begin{proof}
Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.
In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$. Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$. It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$. Hence each
component of $M_P$ is a cycle.
\end{proof}
Let $T$ be a medial tire region. We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.
The following lemma is the basic conservation principle.
\begin{lemma}[Kempe-cycle conservation across level cycles]
\label{lem:kempe-conservation}
Let $C$ be a level cycle of $M$ separating a parent side from a child
side. Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
of $C$ without also leaving it.
Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}
\begin{proof}
By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side. Consider the intersection of $K$ with one of these
regions. Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$. Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
We now use these Kempe cycles to single out the colourings of a full
medial tire graph that respect the annular tooth structure.
\begin{definition}[Kempe-balanced colouring]
\label{def:kempe-balanced}
Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
the subgraph induced by the vertices of colours $a$ and $b$. Since
$\mathsf{M}(T)$ need not be $4$-regular, the components of
$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly
one $P$-Kempe chain.
A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
face of $B(T)$ that is not a tooth---namely the root face or a bite
inner-gap face of Remark~\ref{rem:bite-face-count}. The \emph{tooth
apexes incident to} a valid face $F$ are:
\begin{itemize}
\item the up-tooth apexes (Definition~\ref{def:annular-teeth}), when
$F$ is the outer face;
\item the singleton down-tooth apexes whose annular edge lies on $F$,
when $F$ is interior---the apex on annular edge $m$ being incident to
the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
is none.
\end{itemize}
Bite apexes are never incident to a valid face in this sense.
For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
lie on a $P$-Kempe chain. The colouring $\varphi$ is
\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
every colour pair $P$.
\end{definition}
\begin{remark}[Necessity of Kempe-balance]
\label{rem:kempe-balance-necessary}
A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
$3$-colouring of the whole medial graph $M(G)$ only when it is
Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
condition at some valid face and colour pair cannot extend to a proper
$3$-colouring of $M(G)$. This is an instance of Kempe-cycle
conservation (Lemma~\ref{lem:kempe-conservation}). The tooth apexes
incident to a valid face are boundary medial vertices
(Definition~\ref{def:boundary-medial-vertices}) lying on a single level
cycle of the tire decomposition: the up-tooth apexes lie on the outer
level cycle, and the singleton down-tooth apexes incident to an interior
non-tooth face lie on the inner level cycle bounding that face. In the
$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
meets each level cycle in an even number of $P$-coloured incidences; for
a given valid face these incidences are exactly its incident tooth
apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
This argument is verified computationally. For bite-free pieces---capped
triangulated annuli on annular cycles of length $6,8,10,12$---every proper
$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring. The same
holds for pieces carrying a bite, including the case where singleton down
teeth lie in the bite's inner-gap face: there the inner level cycle splits
into a child level cycle per gap, and conservation across each child cycle
supplies the parity (in the checked example the three singleton down apexes
of a bite gap are a rainbow in every restriction).
\end{remark}
More generally, let $T$ be a medial tire region with boundary
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.
\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$. Let
\[
\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
\item a boundary colouring
\[
\alpha:V(\mathcal C(T))\to\{1,2,3\};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
a pairing $\pi_P$ of the $P$-coloured boundary incidences of
$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}
Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
\kappa_T(\varphi).
\]
\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
\mathcal K_T
=
\left\{
\kappa_T(\varphi):
\varphi \text{ is a proper }3\text{-colouring of } M(T)
\right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}
The annular structure of a tire is useful in two distinct ways. First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices. Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire. Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs. The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.
\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.
Let
\[
\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
\in \mathcal K_T
\]
and
\[
\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
\in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
\item the boundary colourings agree on $C$:
\[
\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
$P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}
In these terms, gluing local colourings is not merely a matter of
matching boundary colours. The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle. The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.
For a tire with one outer level cycle and several inner level cycles,
\[
\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles. The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously. Thus one should view a medial tire as a
multi-output transition operator
\[
\mathcal K_T:
C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.
The guiding principle is therefore:
\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}
\begin{thebibliography}{9}
\bibitem{bauerfeld-nested-tire-decompositions}
@@ -997,6 +666,11 @@ E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-medial-pigeonhole}
E.~Bauerfeld,
\emph{The Medial Pigeonhole Programme},
manuscript (math-research repository), 2026.
\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},