\relax \citation{parent} \citation{parent} \citation{parent} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Setup and background}}{1}{}\protected@file@percent } \newlabel{sec:background}{{1}{1}} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The iterated reduction}}{1}{}\protected@file@percent } \newlabel{sec:iterated-reduction}{{2}{1}} \newlabel{alg:iterated-reduction}{{2.1}{1}} \citation{parent} \citation{parent} \citation{parent} \citation{parent} \citation{parent} \newlabel{rem:alg-invariants}{{2.2}{2}} \newlabel{rem:alg-chord-apex}{{2.3}{2}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Structural lemmas on the algorithm's output}}{2}{}\protected@file@percent } \newlabel{sec:structural-lemmas}{{3}{2}} \newlabel{lem:exactly-one-match}{{3.1}{2}} \citation{parent} \citation{parent} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Algorithm\nonbreakingspace 2.1\hbox {} on $G'=\mathrm {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex and Kempe-cycle conditions of\nonbreakingspace \cite {parent}, found by \texttt {experiments/search\_kempe\_property.py}. \emph {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{3}{}\protected@file@percent } \newlabel{fig:iterated-reduction-trace}{{1}{3}} \newlabel{lem:all-distinct-exists}{{3.2}{3}} \citation{parent} \citation{parent} \bibcite{parent}{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}}{4}{}\protected@file@percent } \gdef \@abspage@last{4}