30 lines
3.3 KiB
TeX
30 lines
3.3 KiB
TeX
\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}
|