didericis
c64c720e5a
Add a --whole mode to draw_medial_tire_cut.py that renders the entire medial graph M(G) (the assembled cut graph), on a Kamada-Kawai layout, with the recognised tires highlighted (black annular vertices, blue/red teeth carrying walk depths, larger red bite apex) and the rest of M(G) in grey. Add the resulting figure (Figure 3) and a describing paragraph to the paper for the n=20 seed-72 example, via an \input-ed .tikz file. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
30 lines
3.1 KiB
TeX
30 lines
3.1 KiB
TeX
\relax
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\citation{bauerfeld-medial-tire}
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\citation{bauerfeld-medial-tire}
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\citation{bauerfeld-medial-tire}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Cutting a full medial tire graph}}{1}{}\protected@file@percent }
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\newlabel{def:walk-depth-cut}{{2.1}{1}}
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\citation{bauerfeld-medial-tire}
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\citation{bauerfeld-medial-tire}
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\newlabel{rem:closing-tooth}{{2.2}{2}}
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\newlabel{ex:worked-cut}{{2.3}{2}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Chaining across the tire tree}}{2}{}\protected@file@percent }
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\citation{bauerfeld-medial-tire}
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\@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 }
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\newlabel{fig:worked-cut}{{1}{3}}
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\newlabel{rem:chaining-candidates}{{3.1}{3}}
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\newlabel{ex:real-cut}{{3.2}{3}}
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\bibcite{bauerfeld-medial-tire}{1}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{12.7778pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\@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 }
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\newlabel{fig:real-cut}{{2}{4}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The whole medial graph $M(G)$ of the random maximal planar graph on $20$ vertices from Example\nonbreakingspace 3.2\hbox {}, with all tire cuts applied. Grey vertices are medial vertices outside any recognised tire; the highlighted tread $T_2$ (cf.\ Figure\nonbreakingspace 2\hbox {}) has black annular medial vertices, blue up-tooth and red down-tooth apexes carrying their walk depths, and the larger red vertex is the bite apex. Drawn by \texttt {experiments/draw\_medial\_tire\_cut.py} with the \texttt {--whole} option.}}{5}{}\protected@file@percent }
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\newlabel{fig:whole-medial}{{3}{5}}
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\gdef \@abspage@last{5}
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