Draft quadrilateral sequencing section
Extend the deep embedding to include the outer face, decompose into quadrilaterals via level-edge pairing on the sphere, and define a deterministic sequence built from four moves (anchor drop, level add, join, ring completion) with a recursive lex-smallest tiebreak on the initial quadrilateral. Attempt the termination theorem and the per-move case analyses. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -1,11 +0,0 @@
|
||||
\relax
|
||||
\citation{baker1994}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent }
|
||||
\bibcite{baker1994}{1}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{2}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{2}
|
||||
@@ -1,54 +0,0 @@
|
||||
# Fdb version 4
|
||||
["pdflatex"] 1777106635.82554 "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" "paper.pdf" "paper" 1777106636.10525 0
|
||||
"/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" 1777106635.61829 7628 2263cb80db1ec8aa25bb60c5652b01c2 ""
|
||||
"/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1 5523663 ec1f96d89b308e150332b305019a3402 ""
|
||||
"/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/pdftex/pdflatex.fmt" 1 3600504 177ced77725200f4fa24b79427ded12f ""
|
||||
"/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/texmf.cnf" 1 44455 00ca67f5a06c9c23b32559f3f48cb4e9 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/map/fontname/texfonts.map" 1 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1 1004 54797486969f23fa377b128694d548df ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1 916 f87d7c45f9c908e672703b83b72241a3 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1 924 9904cf1d39e9767e7a3622f2a125a565 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1 928 2dc8d444221b7a635bb58038579b861a ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1 908 2921f8a10601f252058503cc6570e581 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1 940 75ac932a52f80982a9f8ea75d03a34cf ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1 940 228d6584342e91276bf566bcf9716b83 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmbx10.tfm" 1 1328 c834bbb027764024c09d3d2bf908b5f0 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmcsc10.tfm" 1 1300 63a6111ee6274895728663cf4b4e7e81 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmmi6.tfm" 1 1512 f21f83efb36853c0b70002322c1ab3ad ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmmi8.tfm" 1 1520 eccf95517727cb11801f4f1aee3a21b4 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmr6.tfm" 1 1300 b62933e007d01cfd073f79b963c01526 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmr8.tfm" 1 1292 21c1c5bfeaebccffdb478fd231a0997d ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmsy6.tfm" 1 1116 933a60c408fc0a863a92debe84b2d294 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmsy8.tfm" 1 1120 8b7d695260f3cff42e636090a8002094 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmti10.tfm" 1 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmti8.tfm" 1 1504 1747189e0441d1c18f3ea56fafc1c480 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1 34811 78b52f49e893bcba91bd7581cdc144c0 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1 32001 6aeea3afe875097b1eb0da29acd61e28 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1 36281 c355509802a035cadc5f15869451dcee ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1 35752 024fb6c41858982481f6968b5fc26508 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1 32762 224316ccc9ad3ca0423a14971cfa7fc1 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1 32569 5e5ddc8df908dea60932f3c484a54c0d ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1 32716 08e384dc442464e7285e891af9f45947 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1 37944 359e864bd06cde3b1cf57bb20757fb06 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1 35660 fb24af7afbadb71801619f1415838111 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1 31764 459c573c03a4949a528c2cc7f557e217 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amscls/amsart.cls" 1 61881 a7369c346c2922a758ae6283cc1ed014 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/amsfonts.sty" 1 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd" 1 961 6518c6525a34feb5e8250ffa91731cff ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsb.fd" 1 961 d02606146ba5601b5645f987c92e6193 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsbsy.sty" 1 2222 27db7d52163edae53881b71ff62e754e ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsgen.sty" 1 4173 1b3e76addfb8afcb47db4811d66e1dc6 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsmath.sty" 1 88401 0c3d1897569ad77cb9d8fb25b0bdf668 ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty" 1 4474 c510a88aa5f51b8c773b50a7ee92befd ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amstext.sty" 1 2444 9983e1d0683f102e3b190c64a49313aa ""
|
||||
"/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def" 1 30351 a2b09edc6c93a742566b222c33d0278e ""
|
||||
"paper.aux" 1777106636.0643 429 e867892bc6d7fe276fbd19bcc4f1bc53 "pdflatex"
|
||||
"paper.tex" 1777106635.61829 7628 2263cb80db1ec8aa25bb60c5652b01c2 ""
|
||||
(generated)
|
||||
"paper.aux"
|
||||
"paper.log"
|
||||
"paper.pdf"
|
||||
(rewritten before read)
|
||||
@@ -1,75 +0,0 @@
|
||||
PWD /home/didericis/Code/math-research/papers/plane_depth_sequencing
|
||||
INPUT /nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/texmf.cnf
|
||||
INPUT /nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/pdftex/pdflatex.fmt
|
||||
INPUT /home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex
|
||||
OUTPUT paper.log
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amscls/amsart.cls
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amscls/amsart.cls
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/amsfonts.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/amsfonts.sty
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
INPUT ./paper.aux
|
||||
INPUT ./paper.aux
|
||||
INPUT paper.aux
|
||||
OUTPUT paper.aux
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/map/fontname/texfonts.map
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmr8.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmr6.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmmi8.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmmi6.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmsy8.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmsy6.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam5.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsb.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsb.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsb.fd
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmcsc10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmti8.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmbx10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmcsc10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/tfm/public/cm/cmti10.tfm
|
||||
OUTPUT paper.pdf
|
||||
INPUT /nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map
|
||||
INPUT paper.aux
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmmi7.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr10.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr7.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmti10.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmti8.pfb
|
||||
INPUT /nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/symbols/msam10.pfb
|
||||
@@ -1,168 +0,0 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 25 APR 2026 04:43
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex
|
||||
(/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex
|
||||
LaTeX2e <2025-06-01> patch level 1
|
||||
L3 programming layer <2025-06-09>
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amscls/amsart.cls
|
||||
Document Class: amsart 2020/05/29 v2.20.6
|
||||
\linespacing=\dimen148
|
||||
\normalparindent=\dimen149
|
||||
\normaltopskip=\skip49
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsmath.sty
|
||||
Package: amsmath 2025/06/16 v2.17y AMS math features
|
||||
\@mathmargin=\skip50
|
||||
|
||||
For additional information on amsmath, use the `?' option.
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amstext.sty
|
||||
Package: amstext 2024/11/17 v2.01 AMS text
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsgen.sty
|
||||
File: amsgen.sty 1999/11/30 v2.0 generic functions
|
||||
\@emptytoks=\toks17
|
||||
\ex@=\dimen150
|
||||
)) (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsbsy.sty
|
||||
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
|
||||
\pmbraise@=\dimen151
|
||||
) (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty
|
||||
Package: amsopn 2022/04/08 v2.04 operator names
|
||||
)
|
||||
\inf@bad=\count275
|
||||
LaTeX Info: Redefining \frac on input line 233.
|
||||
\uproot@=\count276
|
||||
\leftroot@=\count277
|
||||
LaTeX Info: Redefining \overline on input line 398.
|
||||
LaTeX Info: Redefining \colon on input line 409.
|
||||
\classnum@=\count278
|
||||
\DOTSCASE@=\count279
|
||||
LaTeX Info: Redefining \ldots on input line 495.
|
||||
LaTeX Info: Redefining \dots on input line 498.
|
||||
LaTeX Info: Redefining \cdots on input line 619.
|
||||
\Mathstrutbox@=\box53
|
||||
\strutbox@=\box54
|
||||
LaTeX Info: Redefining \big on input line 721.
|
||||
LaTeX Info: Redefining \Big on input line 722.
|
||||
LaTeX Info: Redefining \bigg on input line 723.
|
||||
LaTeX Info: Redefining \Bigg on input line 724.
|
||||
\big@size=\dimen152
|
||||
LaTeX Font Info: Redeclaring font encoding OML on input line 742.
|
||||
LaTeX Font Info: Redeclaring font encoding OMS on input line 743.
|
||||
\macc@depth=\count280
|
||||
LaTeX Info: Redefining \bmod on input line 904.
|
||||
LaTeX Info: Redefining \pmod on input line 909.
|
||||
LaTeX Info: Redefining \smash on input line 939.
|
||||
LaTeX Info: Redefining \relbar on input line 969.
|
||||
LaTeX Info: Redefining \Relbar on input line 970.
|
||||
\c@MaxMatrixCols=\count281
|
||||
\dotsspace@=\muskip17
|
||||
\c@parentequation=\count282
|
||||
\dspbrk@lvl=\count283
|
||||
\tag@help=\toks18
|
||||
\row@=\count284
|
||||
\column@=\count285
|
||||
\maxfields@=\count286
|
||||
\andhelp@=\toks19
|
||||
\eqnshift@=\dimen153
|
||||
\alignsep@=\dimen154
|
||||
\tagshift@=\dimen155
|
||||
\tagwidth@=\dimen156
|
||||
\totwidth@=\dimen157
|
||||
\lineht@=\dimen158
|
||||
\@envbody=\toks20
|
||||
\multlinegap=\skip51
|
||||
\multlinetaggap=\skip52
|
||||
\mathdisplay@stack=\toks21
|
||||
LaTeX Info: Redefining \[ on input line 2949.
|
||||
LaTeX Info: Redefining \] on input line 2950.
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 397.
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
) (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/amsfonts.sty
|
||||
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
|
||||
\symAMSa=\mathgroup4
|
||||
\symAMSb=\mathgroup5
|
||||
LaTeX Font Info: Redeclaring math symbol \hbar on input line 98.
|
||||
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
|
||||
)
|
||||
\copyins=\insert252
|
||||
\abstractbox=\box55
|
||||
\listisep=\skip53
|
||||
\c@part=\count287
|
||||
\c@section=\count288
|
||||
\c@subsection=\count289
|
||||
\c@subsubsection=\count290
|
||||
\c@paragraph=\count291
|
||||
\c@subparagraph=\count292
|
||||
\c@figure=\count293
|
||||
\c@table=\count294
|
||||
\abovecaptionskip=\skip54
|
||||
\belowcaptionskip=\skip55
|
||||
\captionindent=\dimen159
|
||||
\thm@style=\toks22
|
||||
\thm@bodyfont=\toks23
|
||||
\thm@headfont=\toks24
|
||||
\thm@notefont=\toks25
|
||||
\thm@headpunct=\toks26
|
||||
\thm@preskip=\skip56
|
||||
\thm@postskip=\skip57
|
||||
\thm@headsep=\skip58
|
||||
\dth@everypar=\toks27
|
||||
)
|
||||
\c@theorem=\count295
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
File: l3backend-pdftex.def 2025-06-09 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count296
|
||||
) (./paper.aux)
|
||||
\openout1 = `paper.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 52.
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 52.
|
||||
(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsb.fd
|
||||
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
)
|
||||
|
||||
[1{/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
|
||||
[2] (./paper.aux)
|
||||
***********
|
||||
LaTeX2e <2025-06-01> patch level 1
|
||||
L3 programming layer <2025-06-09>
|
||||
***********
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
1769 strings out of 467888
|
||||
26102 string characters out of 5405403
|
||||
437018 words of memory out of 5000000
|
||||
30199 multiletter control sequences out of 15000+600000
|
||||
633232 words of font info for 65 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
71i,6n,79p,816b,236s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr10.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr7.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmti10.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/cm/cmti8.pfb></nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (2 pages, 158166 bytes).
|
||||
PDF statistics:
|
||||
71 PDF objects out of 1000 (max. 8388607)
|
||||
42 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
1 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@@ -12,9 +12,9 @@
|
||||
%% (800) 321-4267 (USA and Canada only)
|
||||
%% fax: (401) 331-3842
|
||||
%% email: tech-support@ams.org
|
||||
%%
|
||||
%%
|
||||
%% Copyright 2008-2010, 2014 American Mathematical Society.
|
||||
%%
|
||||
%%
|
||||
%% This work may be distributed and/or modified under the
|
||||
%% conditions of the LaTeX Project Public License, either version 1.3c
|
||||
%% of this license or (at your option) any later version.
|
||||
@@ -22,9 +22,9 @@
|
||||
%% http://www.latex-project.org/lppl.txt
|
||||
%% and version 1.3c or later is part of all distributions of LaTeX
|
||||
%% version 2005/12/01 or later.
|
||||
%%
|
||||
%%
|
||||
%% This work has the LPPL maintenance status `maintained'.
|
||||
%%
|
||||
%%
|
||||
%% The Current Maintainer of this work is the American Mathematical
|
||||
%% Society.
|
||||
%%
|
||||
@@ -119,11 +119,7 @@ Now let $v \in V_d$. Since $\mathrm{depth}(v) = d \geq 1$, there exists $u \in V
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}
|
||||
Let $G$ be a maximal planar graph with a plane embedding and outer cycle $C$. The \emph{deep embedding} of $G$ is the graph $G'$ obtained from $G$ by the following operation: for every 3-cycle $\{u, v, w\} \subseteq V(G)$ such that
|
||||
\[
|
||||
\mathrm{depth}(u) = \mathrm{depth}(v) = \mathrm{depth}(w),
|
||||
\]
|
||||
add a new vertex $x$ to $G$ adjacent to each of $u$, $v$, and $w$.
|
||||
Let $G$ be a maximal planar graph with a plane embedding and outer cycle $C$. The \emph{deep embedding} of $G$ is the graph $G'$ obtained from $G$ by the following operation: for every neutral triangular face $\{u, v, w\}$ of $G$ --- \emph{including the outer face}, whose vertices are the three vertices of $C$ --- add a new vertex $x$ placed in that face and adjacent to each of $u$, $v$, and $w$. The vertex added inside the outer face is denoted $x^*$ and called the \emph{outer-cap vertex}; the three triangular faces it induces with the edges of $C$ are the \emph{outer-cap faces}. We henceforth view $G'$ as embedded on the sphere $S^2$, with no distinguished outer face.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
@@ -139,11 +135,170 @@ We now consider each case under the deep embedding.
|
||||
|
||||
\textit{Case 1: up triangle or down triangle.} These triangles are not modified by the deep embedding, so they remain as faces of $G'$, satisfying the lemma.
|
||||
|
||||
\textit{Case 2: neutral triangle.} The deep embedding inserts a new vertex $x$ adjacent to $u$, $v$, and $w$, replacing the face $\{u,v,w\}$ with three new faces $\{u,v,x\}$, $\{v,w,x\}$, and $\{u,w,x\}$. It remains to determine the depth of $x$ in $G'$. Since $x$ is adjacent only to $u$, $v$, and $w$, every path in $G'$ from $x$ to $C$ must pass through one of them, so $x$ has strictly greater depth than $u$, $v$, and $w$. Each of the three new faces is thus a down triangle, satisfying the lemma.
|
||||
\textit{Case 2: neutral triangle.} The deep embedding inserts a new vertex $x$ adjacent to $u$, $v$, and $w$, replacing the face $\{u,v,w\}$ with three new faces $\{u,v,x\}$, $\{v,w,x\}$, and $\{u,w,x\}$. It remains to determine the depth of $x$ in $G'$. Since $x$ is adjacent only to $u$, $v$, and $w$, every path in $G'$ from $x$ to $C$ must pass through one of them, so $x$ has strictly greater depth than $u$, $v$, and $w$. Each of the three new faces is thus a down triangle, satisfying the lemma. The same argument applies to the outer face: the outer-cap vertex $x^*$ is adjacent to all three vertices of $C$ (which lie at depth $0$), so $\mathrm{depth}(x^*) = 1$, and each of the three outer-cap faces is a down triangle.
|
||||
|
||||
Since every face of $G'$ falls into one of these cases, the result follows.
|
||||
\end{proof}
|
||||
|
||||
\section{Quadrilateral sequencing}
|
||||
|
||||
We now decompose the deep embedding into quadrilaterals by removing level edges, and define a deterministic sequence in which those quadrilaterals are visited.
|
||||
|
||||
\begin{lemma}
|
||||
Every interior face of $G'$ has exactly one level edge.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
By the previous lemma, each interior face is an up triangle (depths $\{d, d+1, d+1\}$) or a down triangle (depths $\{d, d, d+1\}$). In both cases, exactly one of the three vertex pairs has equal depth.
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}
|
||||
Let $e = \{p, q\}$ be any level edge of $G'$. Then $e$ is the unique level edge of both faces incident to it.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
On the sphere, both faces $T, T'$ incident to $e$ are triangles. Since $p$ and $q$ have equal depth, $e$ is a level edge of $T$ and of $T'$, and by the previous lemma each has $e$ as its unique level edge.
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}
|
||||
The \emph{quadrilateral decomposition} of $G'$ pairs each face of $G'$ with the face on the other side of its (unique) level edge. Each pair, together with the four non-level edges of the two triangles, bounds a \emph{quadrilateral} of the decomposition.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
Because $G'$ is taken on the sphere, every edge lies between two triangular faces, so the pairing above applies uniformly. In particular, each edge of $C$ is a level edge shared between one interior boundary down triangle (depths $\{0, 0, 1\}$, with the depth-$1$ vertex inside $C$) and one outer-cap down triangle (depths $\{0, 0, 1\}$, with apex $x^*$). The three resulting quadrilaterals, one per edge of $C$, are the \emph{boundary deep diamonds}; they are the outermost quadrilaterals of the decomposition.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}
|
||||
Each quadrilateral is one of three types, classified by the depths of its two non-level vertices relative to the depth $d$ of the shared level edge:
|
||||
\begin{itemize}
|
||||
\item a \emph{shallow diamond}, formed by two up triangles, with vertex depths $(d-1, d, d-1, d)$ around the boundary;
|
||||
\item a \emph{deep diamond}, formed by two down triangles, with vertex depths $(d+1, d, d+1, d)$ around the boundary;
|
||||
\item an \emph{S quad}, formed by one up and one down triangle, with vertex depths $(d-1, d, d+1, d)$ around the boundary.
|
||||
\end{itemize}
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
For the remainder of this section, fix a plane embedding of $G'$ by designating one of the three outer-cap faces as the outer face of the plane drawing; the outer cycle of this plane embedding is the boundary of that designated outer-cap face. Orient this outer cycle counterclockwise so that the remaining faces of $G'$ lie to its left. This induces a canonical cyclic order on the edges incident to each vertex and a notion of \emph{left} and \emph{right} on each triangle.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}
|
||||
A \emph{slice} of $G'$ is a connected, simply connected region of the plane formed by the union of a subset of quadrilaterals from the decomposition, together with its closed boundary walk in $G'$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Move code]
|
||||
Each move is assigned a numerical code: anchor drop $= 0$, level add $= 1$, join $= 2$, ring completion $= 3$. Given a depth sequence $Q_1, Q_2, \ldots, Q_N$, its \emph{move-code string} is the word $m_2 m_3 \cdots m_N \in \{0, 1, 2, 3\}^{N-1}$, where $m_n$ is the code of the move that produced $Q_n$ from $S_{n-1}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Initial quad]
|
||||
The depth sequence begins by choosing as $Q_1$ the boundary deep diamond whose resulting move-code string is lexicographically smallest among the three boundary deep diamonds. If multiple boundary deep diamonds yield the same move-code string, $Q_1$ is not uniquely determined; this case is called a \emph{rotational tie} and corresponds to a symmetry of $G'$ that permutes the boundary deep diamonds. The initial slice is $S_1 = Q_1$.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
The tiebreak is recursive: the choice of $Q_1$ depends on the move-code strings produced by each of the three candidate starts, which in turn depend on the entire sequence each candidate produces. Equivalently, run the deterministic sequencing scheme from each of the three boundary deep diamonds and compare the resulting move-code strings; pick the start that produces the lexicographically smallest string.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Anchor drop]
|
||||
Suppose a slice $S_n$ has been constructed and the lower-rightmost portion of its boundary (in the fixed plane embedding) is a down triangle $a$ whose right edge $e$ is exposed. If there exists an S quad $Q \notin S_n$ whose up triangle $b$ has $e$ as its left edge, then the \emph{anchor drop} sets
|
||||
\[
|
||||
Q_{n+1} = Q, \qquad S_{n+1} = S_n \cup Q.
|
||||
\]
|
||||
The anchor drop introduces two new vertices: the second endpoint of $b$'s level edge (at depth one greater than $a$'s level edge) and the apex of $Q$'s down triangle (at depth two greater).
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Level add]
|
||||
Suppose a slice $S_n$ has been constructed. Consider the quadrilaterals $Q \notin S_n$ such that exactly three of the four vertices of $Q$ lie on the right boundary of $S_n$. Among these, the \emph{level add} chooses the $Q$ whose attachment to the right boundary occurs at the bottommost position (i.e., the last position encountered when scanning the right boundary from top to bottom). Then
|
||||
\[
|
||||
Q_{n+1} = Q, \qquad S_{n+1} = S_n \cup Q.
|
||||
\]
|
||||
By construction, the level add introduces exactly one new vertex.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Join]
|
||||
Suppose a slice $S_n$ has been constructed. Consider the deep diamonds $Q \notin S_n$ such that one of the two down triangles comprising $Q$ shares an edge with the right boundary of $S_n$ (so two of $Q$'s four vertices lie on the right boundary). Among these, the \emph{join} chooses the $Q$ whose shared boundary edge is the bottommost in the right boundary scanned from top to bottom. Then
|
||||
\[
|
||||
Q_{n+1} = Q, \qquad S_{n+1} = S_n \cup Q.
|
||||
\]
|
||||
Generically, the join introduces two new vertices: the second endpoint of $Q$'s level edge (depth $d$) and the apex of $Q$'s second down triangle (depth $d+1$), where the shared boundary edge has depths $\{d, d+1\}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
Like the anchor drop, the join generically introduces two new vertices. Unlike the anchor drop, neither new vertex is at greater depth than the existing slice: the join extends the slice horizontally along the depth-$d$ and depth-$(d+1)$ rings rather than descending one level deeper.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Ring completion]
|
||||
Suppose a slice $S_n$ has been constructed. Consider the quadrilaterals $Q \notin S_n$ such that all four vertices of $Q$ are already vertices of $S_n$. Among these, the \emph{ring completion} chooses the $Q$ whose attachment to the right boundary of $S_n$ is bottommost (i.e., the last attachment encountered when scanning the right boundary from top to bottom). Then
|
||||
\[
|
||||
Q_{n+1} = Q, \qquad S_{n+1} = S_n \cup Q.
|
||||
\]
|
||||
By construction, the ring completion introduces no new vertices.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Move selection]
|
||||
At each step $n \geq 1$, the next quadrilateral $Q_{n+1}$ is chosen by the first applicable move in the following order of precedence:
|
||||
\begin{enumerate}
|
||||
\item anchor drop;
|
||||
\item level add;
|
||||
\item join;
|
||||
\item ring completion.
|
||||
\end{enumerate}
|
||||
That is, each move is consulted only when no higher-precedence move applies.
|
||||
\end{definition}
|
||||
|
||||
Let $N$ denote the total number of quadrilaterals in the decomposition of $G'$; equivalently, $N = |F(G')|/2$, where $F(G')$ is the set of triangular faces of $G'$.
|
||||
|
||||
\begin{theorem}[Termination and coverage]
|
||||
The sequence $Q_1, Q_2, \ldots$ generated by repeatedly applying the move-selection rule starting from any choice of initial quadrilateral $Q_1$ terminates after exactly $N$ steps. Moreover, every quadrilateral of the decomposition appears in the sequence exactly once, and $S_N = G'$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
We prove the three claims comprising the theorem.
|
||||
|
||||
\emph{(1) No quadrilateral is visited twice.} By construction, each move chooses $Q_{n+1} \notin S_n$ and sets $S_{n+1} = S_n \cup Q_{n+1}$. Hence $n \mapsto Q_n$ is injective.
|
||||
|
||||
\emph{(2) Each $S_n$ is a slice.} We proceed by induction on $n$.
|
||||
|
||||
\emph{Base case.} $S_1 = Q_1$ is a single quadrilateral, hence a closed topological disk on the sphere with boundary the closed walk along $Q_1$'s four perimeter edges.
|
||||
|
||||
\emph{Inductive step.} Suppose $S_n$ is a slice. We show that for each of the four moves, $S_{n+1} = S_n \cup Q_{n+1}$ is again a slice. Topologically, $S_n$ is a closed disk and $Q_{n+1}$ is a closed disk; their union is a closed disk iff their intersection $S_n \cap Q_{n+1}$ is a connected arc on each of their boundaries. We verify this in each case below by identifying the intersection precisely.
|
||||
|
||||
\emph{(2.1) Anchor drop.} The new quadrilateral $Q_{n+1}$ shares exactly one edge $e$ with $S_n$: the right edge of the boundary down triangle $a$, which is also the left edge of the up triangle $b$. The intersection $S_n \cap Q_{n+1} = e$ is a single edge, a connected arc.
|
||||
|
||||
\emph{(2.2) Level add.} Three vertices $v_1, v_2, v_3$ of $Q_{n+1}$ lie on the right boundary of $S_n$. By the move's precedence, no anchor drop or higher-priority move applied at step $n+1$. We claim the two perimeter edges of $Q_{n+1}$ connecting these three vertices, namely $\{v_1, v_2\}$ and $\{v_2, v_3\}$, lie on the boundary of $S_n$. Suppose otherwise: then at least one of these edges has both adjacent faces in $S_n^c$, which means the face of $G'$ on the side of that edge opposite $Q_{n+1}$ is also outside $S_n$. By the structure of the move precedence and the inductive assumption, all such configurations would have been resolved by a higher-priority move first; hence at the moment level add fires, the only viable attachment configuration is the 2-edge path $\{v_1, v_2, v_3\}$. The intersection $S_n \cap Q_{n+1}$ is therefore this 2-edge path, a connected arc.
|
||||
|
||||
\emph{(2.3) Join.} The deep diamond $Q_{n+1}$ has one of its two down triangles sharing an edge $e$ with the right boundary of $S_n$. The intersection $S_n \cap Q_{n+1} = e$, a single edge, is a connected arc. (The second down triangle of $Q_{n+1}$ contributes two new vertices and lies entirely in $S_n^c$.)
|
||||
|
||||
\emph{(2.4) Ring completion.} All four vertices of $Q_{n+1}$ lie in $V(S_n)$, and by precedence none of the previous three moves applied. We claim that in this case, at least three of $Q_{n+1}$'s four perimeter edges lie on the boundary of $S_n$, and these edges together form a connected arc along the boundary of $Q_{n+1}$. Indeed, any perimeter edge of $Q_{n+1}$ with both endpoints in $S_n$ but neither side in $S_n$ would force one of the higher-priority moves (specifically join or level add, applied with respect to the face on the other side of that edge); since no such move applied, the configuration is constrained so that the intersection $S_n \cap Q_{n+1}$ is a connected arc consisting of three or four of $Q_{n+1}$'s perimeter edges.
|
||||
|
||||
In all four cases the intersection $S_n \cap Q_{n+1}$ is a connected arc, so $S_{n+1}$ is a closed topological disk and hence a slice. The boundary walk of $S_{n+1}$ is obtained from that of $S_n$ by deleting the arc $S_n \cap Q_{n+1}$ and inserting the complementary arc of $Q_{n+1}$'s boundary.
|
||||
|
||||
\emph{(3) As long as $S_n \subsetneq G'$, some move applies.} Let $S_n$ be a slice with $S_n \subsetneq G'$. The complement $G' \setminus S_n$ is a non-empty closed disk on the sphere whose boundary coincides with the boundary of $S_n$. Pick an edge $e$ of the right boundary of $S_n$, and let $F$ be the face of $G'$ on the side of $e$ opposite $S_n$; then $F$ lies in $S_n^c$. Let $Q$ be the quadrilateral containing $F$.
|
||||
|
||||
Let $k$ be the number of vertices of $Q$ lying in $V(S_n)$, and let $j$ be the number of perimeter edges of $Q$ lying on the boundary of $S_n$. Since the edge $e$ belongs to $Q$ and lies on the boundary of $S_n$, we have $j \geq 1$ and $k \geq 2$.
|
||||
|
||||
\emph{Case $k = 2$, $j = 1$.} Only the two endpoints of $e$ are in $S_n$.
|
||||
\begin{itemize}
|
||||
\item If $Q$ is an S quad and $e$ is the left edge of $Q$'s up triangle, then writing $a$ for the down triangle of $S_n$ across $e$, the anchor drop hypothesis holds and the move applies (possibly with a different choice of $a$ at the lower-rightmost position).
|
||||
\item If $Q$ is a deep diamond, then $e$ belongs to one of its down triangles, and the join hypothesis holds.
|
||||
\item If $Q$ is a shallow diamond, then neither anchor drop nor join applies directly to $Q$. In this case, follow the boundary of $S_n$ to find an adjacent face $F'$ also in $S_n^c$ whose containing quadrilateral $Q'$ admits one of the four moves; such an adjacent quadrilateral exists because $S_n^c$ is connected and is bounded by a closed walk in $G'$.
|
||||
\end{itemize}
|
||||
|
||||
\emph{Case $k = 3$, $j = 2$.} The three boundary vertices form a 2-edge path on $Q$'s outline, and the level add hypothesis holds. (No higher-priority move need have applied; if anchor drop also applies, the rule gives precedence to anchor drop.)
|
||||
|
||||
\emph{Case $k = 4$.} All four vertices of $Q$ are in $V(S_n)$, and $j \geq 1$ by assumption. If a higher-priority hypothesis (anchor drop, level add, join) holds, the corresponding move applies. Otherwise the ring completion hypothesis holds and that move applies.
|
||||
|
||||
In each case, some move adds either $Q$ or an adjacent quadrilateral $Q'$ to the slice, contradicting the assumption that no move applies. Hence as long as $S_n \subsetneq G'$, some move applies and $S_{n+1}$ is well-defined.
|
||||
|
||||
Combining (1)--(3): the sequence strictly grows $|S_n|$ by exactly one quadrilateral per step, never revisits a quadrilateral, and must continue until $S_n = G'$. The total number of steps is therefore exactly $N$.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
Two delicate sub-arguments in the proof of (2) and (3) deserve attention: (a) in (2.2) and (2.4), we relied on the move precedence to rule out configurations where the intersection $S_n \cap Q_{n+1}$ is disconnected; and (b) in (3), the shallow-diamond sub-case argues by ``move to an adjacent face'' but does not pin down which face. A more rigorous treatment would prove the equivalence of the following two statements: (i) the four moves cover every attachment configuration arising from a slice; (ii) at each step the move-selection rule produces a unique well-defined next quadrilateral.
|
||||
\end{remark}
|
||||
|
||||
% TODO: state and prove a uniqueness result that the sequence
|
||||
% $Q_1, \ldots, Q_N$ is completely determined by the choice of $Q_1$ and the
|
||||
% plane embedding.
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{baker1994}
|
||||
|
||||
Reference in New Issue
Block a user