Replace plane_depth_labelling with plane_depth_sequencing paper and script, remove unused lib modules
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
This commit is contained in:
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@@ -1,145 +0,0 @@
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"/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" 1777053383.80427 278 6ee4d5d6b3925666e01309d058e35079 "pdflatex"
|
||||
"paper.tex" 1777053383.36427 4888 bd237dd4af63f4b6efd49e77e2265fe5 ""
|
||||
(generated)
|
||||
"paper.aux"
|
||||
"paper.log"
|
||||
"paper.pdf"
|
||||
(rewritten before read)
|
||||
@@ -0,0 +1,75 @@
|
||||
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
|
||||
@@ -0,0 +1,168 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 24 APR 2026 13:56
|
||||
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.
|
||||
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|
||||
LaTeX Info: Redefining \bmod on input line 904.
|
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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
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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***********
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LaTeX2e <2025-06-01> patch level 1
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%% filename: amsart-template.tex
|
||||
%% version: 1.1
|
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%% date: 2014/07/24
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%%
|
||||
%% American Mathematical Society
|
||||
%% Technical Support
|
||||
%% Publications Technical Group
|
||||
%% 201 Charles Street
|
||||
%% Providence, RI 02904
|
||||
%% USA
|
||||
%% tel: (401) 455-4080
|
||||
%% (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.
|
||||
%% The latest version of this license is in
|
||||
%% http://www.latex-project.org/lppl.txt
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||||
%% and version 1.3c or later is part of all distributions of LaTeX
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%% version 2005/12/01 or later.
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%%
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%% This work has the LPPL maintenance status `maintained'.
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%%
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%% The Current Maintainer of this work is the American Mathematical
|
||||
%% Society.
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||||
%%
|
||||
%% ====================================================================
|
||||
|
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% AMS-LaTeX v.2 template for use with amsart
|
||||
%
|
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% Remove any commented or uncommented macros you do not use.
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\documentclass{amsart}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{example}[theorem]{Example}
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\newtheorem{xca}[theorem]{Exercise}
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\theoremstyle{remark}
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\numberwithin{equation}{section}
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\begin{document}
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\title{Plane Depth Sequencing}
|
||||
|
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% Remove any unused author tags.
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|
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% author one information
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||||
\author{Eric Bauerfeld}
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\address{}
|
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\curraddr{}
|
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\email{}
|
||||
\thanks{}
|
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\subjclass[2010]{Primary }
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\keywords{}
|
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\date{}
|
||||
|
||||
\dedicatory{}
|
||||
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||||
\begin{abstract}
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Definitions}
|
||||
|
||||
\begin{definition}
|
||||
Let $G$ be a graph with a plane embedding, and let $C$ be the outer cycle of that embedding. The \emph{plane depth} of a vertex $v \in V(G)$ relative to the embedding and $C$ is
|
||||
\[
|
||||
\mathrm{depth}(v) = \min_{u \in V(C)} d(v, u),
|
||||
\]
|
||||
where $d(v, u)$ denotes the graph distance between $v$ and $u$ in $G$.
|
||||
\end{definition}
|
||||
|
||||
\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 $w$.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
Let $G'$ be the deep embedding of a maximal planar graph $G$. For each face of $G'$, the plane depths of its three vertices are either $d, d+1, d+1$ or $d, d, d+1$ for some $d \geq 0$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
We first establish that for any edge $\{p, q\}$ in $G$, the depths of $p$ and $q$ differ by at most $1$. Suppose for contradiction that $\mathrm{depth}(p) = d$ and $\mathrm{depth}(q) = d + n$ for some $n \geq 2$. Since $\mathrm{depth}(p) = d$, there exists a path of length $d$ from $p$ to some vertex of $C$. Prepending the edge $\{q, p\}$ gives a path of length $d + 1$ from $q$ to $C$, so $\mathrm{depth}(q) \leq d + 1 < d + n$, a contradiction. The case $\mathrm{depth}(q) = d - n$ is handled identically: there exists a path of length $d - n$ from $q$ to some vertex of $C$, and prepending the edge $\{p, q\}$ gives a path of length $d - n + 1 \leq d - 1 < d$ from $p$ to $C$, contradicting $\mathrm{depth}(p) = d$.
|
||||
|
||||
Since $G$ is a triangulation, every interior face of $G$ is a triangle $\{u,v,w\}$ with all three pairs adjacent. By the above, each pair of vertices in a triangle differs in depth by at most $1$, so no triangle can contain vertices of depths $d$ and $d+2$ simultaneously. The possible depth patterns for a triangle in $G$ are therefore exactly $d,d,d$, or $d,d,d+1$, or $d,d+1,d+1$.
|
||||
|
||||
We now consider each case under the deep embedding.
|
||||
|
||||
\textit{Case 1: depths $d,d,d+1$ or $d,d+1,d+1$.} These triangles are not modified by the deep embedding, so they remain as faces of $G'$ with the stated depth patterns, satisfying the lemma.
|
||||
|
||||
\textit{Case 2: depths $d,d,d$.} 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 thus has depth pattern $d,d,d+1$, satisfying the lemma.
|
||||
|
||||
Since every face of $G'$ falls into one of these cases, the result follows.
|
||||
\end{proof}
|
||||
|
||||
\end{document}
|
||||
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||||
%-----------------------------------------------------------------------
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% End of amsart-template.tex
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%-----------------------------------------------------------------------
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Reference in New Issue
Block a user