\relax 
\@writefile{toc}{\contentsline {paragraph}{The chosen 6-edge cut.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Resulting half-graphs.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Relation to the existing tire framework.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Justification of the per-tire half.}{4}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Justification of the chain half.}{4}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{What this would (and would not) prove.}{4}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Status.}{4}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{6}{}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:cut-depth-label}{{1}{6}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{7}{}\protected@file@percent }
\newlabel{fig:cut-tire}{{2}{7}}
\gdef \@abspage@last{7}
