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\newlabel{fig:level-source}{{1}{1}{The two kinds of level source on a 7-vertex triangulation $T$ (K\textsubscript {4} with vertices $4,5,6$ stacked into the three interior faces). Left: the face source $S = \{0,1,2\}$ (level-0 vertices are the corners of the highlighted triangle). Right: the degree-$3$ vertex source $S = \{4\}$}{figure.1}{}}
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\newlabel{fig:levels}{{2}{2}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level}{figure.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A level cycle in the triangulation of Figure\nonbreakingspace \ref  {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$.}}{2}{figure.3}\protected@file@percent }
\newlabel{fig:level-cycle}{{3}{2}{A level cycle in the triangulation of Figure~\ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$}{figure.3}{}}
\newlabel{def:edge-switch}{{2.4}{2}{Edge switch}{theorem.2.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces An edge switch on the level cycle of Figure\nonbreakingspace \ref  {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$.}}{3}{figure.4}\protected@file@percent }
\newlabel{fig:edge-switch}{{4}{3}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref  {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$.}}{3}{figure.5}\protected@file@percent }
\newlabel{fig:parity-subgraph}{{5}{3}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.5}{}}
\newlabel{def:facial-depth}{{2.6}{3}{Facial depth}{theorem.2.6}{}}
\newlabel{def:surface-switch}{{2.7}{3}{Surface switch}{theorem.2.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Facial depths in a maximal outerplanar graph on $12$ vertices. The six green ear-triangles share an edge with the outer $12$-cycle and so lie in $\mathcal  {B}$ (depth $0$). The three yellow ``in-between'' triangles $(0,2,4),(4,6,8),(0,8,10)$ have only diagonal edges but each is dual-adjacent to ears, giving them $\mathrm  {depth} = 1$. The central triangle $(0,4,8)$ is also all-diagonal; its dual neighbours are the three depth-$1$ triangles, so it is isolated with $\mathrm  {depth} = 2$.}}{4}{figure.6}\protected@file@percent }
\newlabel{fig:facial-depth}{{6}{4}{Facial depths in a maximal outerplanar graph on $12$ vertices. The six green ear-triangles share an edge with the outer $12$-cycle and so lie in $\mathcal {B}$ (depth $0$). The three yellow ``in-between'' triangles $(0,2,4),(4,6,8),(0,8,10)$ have only diagonal edges but each is dual-adjacent to ears, giving them $\mathrm {depth} = 1$. The central triangle $(0,4,8)$ is also all-diagonal; its dual neighbours are the three depth-$1$ triangles, so it is isolated with $\mathrm {depth} = 2$}{figure.6}{}}
\newlabel{def:balanced-surface-switch}{{2.8}{4}{Balanced surface switch}{theorem.2.8}{}}
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\newlabel{thm:outerplanar-component}{{3.1}{4}{}{theorem.3.1}{}}
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\newlabel{prop:balanced-descent}{{3.3}{5}{}{theorem.3.3}{}}
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\newlabel{ex:preprocessing}{{3.5}{6}{}{theorem.3.5}{}}
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\newlabel{fig:no-balanced}{{7}{7}{$9$-vertex maximal outerplanar $L_k$. $F = (0,3,6)$ has $\mathrm {depth} = 1$ and all three of its edges have span $2$, so none of $F$'s depth-$0$ neighbours is an ear. No balanced surface switch is available on $F$}{figure.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces One step of preprocessing on the $9$-vertex example. Left: $F = (0,3,6)$ has no edge of span $1$; the chosen surface-switch edge $uv = 03$ (red) is unbalanced. Right: after the switch $03 \DOTSB \mapstochar \rightarrow 26$ (green), the new depth-$1$ face $A = (0,2,6)$ has its edge $02$ (red) at span $1$, exposing the ear $(0,1,2)$ as a balanced surface-switch target.}}{7}{figure.8}\protected@file@percent }
\newlabel{fig:preprocessing}{{8}{7}{One step of preprocessing on the $9$-vertex example. Left: $F = (0,3,6)$ has no edge of span $1$; the chosen surface-switch edge $uv = 03$ (red) is unbalanced. Right: after the switch $03 \mapsto 26$ (green), the new depth-$1$ face $A = (0,2,6)$ has its edge $02$ (red) at span $1$, exposing the ear $(0,1,2)$ as a balanced surface-switch target}{figure.8}{}}
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\newlabel{fig:d2-recursive}{{9}{8}{Recursive lopsidedness at $d = 2$. Left: $F = (0,8,16)$ depth $2$, every arm doubly-lopsided. Middle: one preprocessing switch $(0,8) \mapsto (2,16)$ exposes the first lopsided layer; the new depth-$2$ face $(2,8,16)$ still has no balanced switch. Right: a second preprocessing switch $(8,2) \mapsto (4,16)$ reaches the inner balanced face $K_0 = (4,6,8)$, whose two non-$F$ neighbours are both ears; the depth-$2$ face $(4,8,16)$ now admits a balanced surface switch on edge $(4,8)$}{figure.9}{}}
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\citation{sleator-tarjan-thurston}
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\newlabel{thm:reachability}{{4.1}{10}{}{theorem.4.1}{}}
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