diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index a0ae3e6..147875f 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -6,38 +6,30 @@ \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\newlabel{def:partial-tire-dual}{{1.1}{1}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent } -\newlabel{fig:partial-tire-dual-example}{{1}{2}} +\citation{bauerfeld-nested-tires} +\newlabel{def:partial-tire-dual}{{1.1}{2}} \newlabel{prop:partial-tire-dual-structure}{{1.2}{2}} -\citation{bauerfeld-nested-tires} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.2\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{3}{}\protected@file@percent } -\newlabel{fig:partial-tire-dual-bridge}{{2}{3}} -\newlabel{prop:no-level-d-pinch}{{1.3}{3}} -\citation{bauerfeld-nested-tires} -\newlabel{lem:tire-component}{{1.4}{4}} -\citation{bauerfeld-depth} -\citation{bauerfeld-nested-tires} -\citation{bauerfeld-depth} -\citation{bauerfeld-nested-tires} -\newlabel{thm:tread-partition}{{1.5}{6}} -\newlabel{rem:tire-component-degenerate}{{1.6}{6}} +\newlabel{prop:edge-vertex-bijection}{{1.3}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{3}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-example}{{1}{3}} +\newlabel{rem:edge-vertex-corollary}{{1.4}{3}} \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} -\newlabel{rem:tire-no-extra-hypotheses}{{1.7}{7}} -\newlabel{prop:edge-vertex-bijection}{{1.8}{7}} -\newlabel{rem:edge-vertex-corollary}{{1.9}{7}} -\newlabel{def:tire-annular-subgraph}{{1.10}{7}} -\newlabel{def:tire-annular-face-connector}{{1.11}{8}} -\newlabel{def:spokes}{{1.12}{8}} -\newlabel{rem:facial-dual-spoke-only}{{1.13}{8}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{8}{}\protected@file@percent } -\newlabel{sec:latin-conjecture}{{2}{8}} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{9}{}\protected@file@percent } -\newlabel{fig:facial-dual-choices}{{3}{9}} -\newlabel{conj:latin}{{2.1}{9}} -\newlabel{conj:chain-latin}{{2.2}{9}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.2\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{4}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-bridge}{{2}{4}} +\newlabel{def:tire-annular-subgraph}{{1.5}{4}} +\newlabel{def:tire-annular-face-connector}{{1.6}{4}} +\newlabel{def:spokes}{{1.7}{5}} +\newlabel{rem:facial-dual-spoke-only}{{1.8}{5}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{5}{}\protected@file@percent } +\newlabel{sec:latin-conjecture}{{2}{5}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{6}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{3}{6}} +\newlabel{conj:latin}{{2.1}{6}} +\newlabel{conj:chain-latin}{{2.2}{6}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tires}{2} \newlabel{tocindent-1}{0pt} @@ -45,5 +37,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } -\gdef \@abspage@last{10} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } +\gdef \@abspage@last{7} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index 88a767b..a3e5946 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.log +++ b/papers/coloring_nested_tire_dual_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:09 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:13 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -195,32 +195,34 @@ e File: fig_partial_tire_dual.png Graphic file (type png) -Package pdftex.def Info: fig_partial_tire_dual.png used on input line 143. +Package pdftex.def Info: fig_partial_tire_dual.png used on input line 151. 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PDF statistics: - 159 PDF objects out of 1000 (max. 8388607) - 94 compressed objects within 1 object stream + 145 PDF objects out of 1000 (max. 8388607) + 85 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 16 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_dual_graphs/paper.pdf b/papers/coloring_nested_tire_dual_graphs/paper.pdf index 28c8f73..eb16180 100644 Binary files a/papers/coloring_nested_tire_dual_graphs/paper.pdf and b/papers/coloring_nested_tire_dual_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_dual_graphs/paper.tex b/papers/coloring_nested_tire_dual_graphs/paper.tex index 2ecc0b4..5264366 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -46,17 +46,17 @@ \begin{abstract} This is a follow-up to \cite{bauerfeld-nested-tires}, which establishes the basic vocabulary of tire graphs $T$ and dual -depth. Building on those definitions, we define the -\emph{partial tire dual} $D(T)$ and analyse its structure in the -spoke-only case (a corona graph $C_{n+m} \circ K_1$), prove the -tire-component lemma -exhibiting every BFS-level component as a tire graph, give an -edge-vertex coloring bijection that reduces counting proper -$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings -of a cycle, and develop the tire-annular-subgraph, face-connector, -and inner/outer-spoke structures in $G'$. A concluding section -records a Latin-substructure conjecture for chain-pigeonhole -compatibility of adjacent tires. +depth, along with the tire-component lemma and the tire-tread +partition theorem. Building on those structural results, we +define the \emph{partial tire dual} $D(T)$ and analyse its +structure in the spoke-only case (a corona graph $C_{n+m} \circ +K_1$), give an edge-vertex coloring bijection that reduces +counting proper $3$-edge-colorings of $D(T)$ to counting proper +$3$-vertex-colorings of a cycle, and develop the +tire-annular-subgraph, face-connector, and inner/outer-spoke +structures in $G'$. A concluding section records a +Latin-substructure conjecture for chain-pigeonhole compatibility +of adjacent tires. \end{abstract} \maketitle @@ -87,7 +87,15 @@ particular we use, without restating, the notions of: with outer/inner boundaries and annular edges (\cite[Definition~1.5]{bauerfeld-nested-tires}); \item face/edge counts - (\cite[Remark~1.6]{bauerfeld-nested-tires}). + (\cite[Remark~1.6]{bauerfeld-nested-tires}); + \item the \emph{tire-component lemma} + (\cite[Lemma~1.8]{bauerfeld-nested-tires}), which exhibits + each connected component of $G'_d$ as a tire graph + whose tire tread is the union of its depth-$d$ faces; + \item the \emph{tire-tread partition theorem} + (\cite[Theorem~1.9]{bauerfeld-nested-tires}), which shows + the tire treads from a level source partition the bounded + faces of $G$. \end{itemize} Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) @@ -209,277 +217,6 @@ cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n + and there are also $n + m$ leaves attached one-per-cycle-vertex. \end{proof} -\begin{proposition}[Source-side simple-cycle property] -\label{prop:no-level-d-pinch} -Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and -single-vertex source $v_0$. Let $d \geq 1$, $v \in L_d$, and let -$C'$ be a connected component of $G'_d$ such that $v$ is incident to -some face in $F_{C'}$. Then the depth-$d$ faces in $F_{C'}$ incident -to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$. - -Equivalently: for any such component, the source-side boundary of -$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$). -\end{proposition} - -\begin{proof} -Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$ -lie in two or more disjoint arcs of $v$'s rotation. Adjacent vertices -in $G$ differ in level by at most $1$, so a face at $v$ has depth -exactly $d$ iff both other vertices have level $\geq d$, and depth -$\leq d-1$ iff at least one has level $d-1$. Hence the gaps between -the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of -$v$, occurring in at least two disjoint arcs of $v$'s rotation. Pick -$p$ in one such gap and $q$ in another. - -The BFS ball $G[L_{d$ sub-regions, the inner outerplanar graph -$O$ captures the multi-hole structure as a disconnected or -non-$2$-connected outerplanar graph (with bridges or multiple -components), and its outer-face boundary closed walk serves as -$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices -multiple times. - -In the special case $d = 0$ with single-vertex source $S = \{v_0\}$, -$R_{C'}$ is the star of $v_0$, a topological closed disk with one -boundary cycle (the link of $v_0$); the corresponding tire graph has -degenerate outer boundary $\{v_0\}$. -\end{remark} \begin{proposition}[Edge--vertex coloring bijection for $D(T)$] \label{prop:edge-vertex-bijection} diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index ac01d65..f7d34bd 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -6,6 +6,16 @@ \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{2}} \newlabel{def:tire-graph}{{1.5}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } +\newlabel{fig:tire-example}{{2}{3}} +\newlabel{rem:tire-counts}{{1.6}{3}} +\newlabel{prop:no-level-d-pinch}{{1.7}{3}} +\newlabel{lem:tire-component}{{1.8}{4}} +\citation{bauerfeld-depth} +\citation{bauerfeld-depth} +\newlabel{thm:tread-partition}{{1.9}{6}} +\newlabel{rem:tire-component-degenerate}{{1.10}{6}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tire-duals}{2} \newlabel{tocindent-1}{0pt} @@ -13,8 +23,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } -\newlabel{fig:tire-example}{{2}{3}} -\newlabel{rem:tire-counts}{{1.6}{3}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent } -\gdef \@abspage@last{3} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } +\gdef \@abspage@last{7} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 0050bb9..09b6a1f 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:01 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:13 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,47 +192,53 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] - + File: fig_dual_depth.png Graphic file (type png) -Package pdftex.def Info: fig_dual_depth.png used on input line 117. +Package pdftex.def Info: fig_dual_depth.png used on input line 122. 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PDF statistics: - 94 PDF objects out of 1000 (max. 8388607) - 54 compressed objects within 1 object stream + 138 PDF objects out of 1000 (max. 8388607) + 82 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index d5f76eb..cd1f1c5 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index f1fc28c..9578dc8 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -44,15 +44,20 @@ \dedicatory{} \begin{abstract} -We establish the foundational definitions for studying the -Four Colour Theorem through nested level-structures on plane -triangulations. A \emph{level source} of a triangulation $G$ -induces a BFS layering of $G$, which in turn endows the inner -planar dual $G'$ with a \emph{dual depth} grading. We isolate the -basic object of study --- the \emph{tire graph} $T$, a plane graph -whose outer and inner boundaries bound the \emph{tire tread} $R$, -a closed region triangulated by the \emph{annular edges} -$E_{\mathrm{ann}}$ --- and record its face/edge counts. +We establish the foundational structure of nested +level-induced tire decompositions of a plane triangulation $G$. +A \emph{level source} of $G$ induces a BFS layering of $G$ and +endows the inner planar dual $G'$ with a \emph{dual depth} +grading. The basic object of study is the \emph{tire graph} +$T$ --- a plane graph whose outer and inner boundaries bound a +closed planar region, the \emph{tire tread} $R$, triangulated by +the \emph{annular edges} $E_{\mathrm{ann}}$. Our main structural +result, the \emph{tire-component lemma}, exhibits each connected +component of $G'_d$ as a tire graph; the \emph{tire-tread +partition theorem} consequence shows the resulting tire treads +partition the bounded faces of $G$. Coloring questions on +$G$ thereby factor through coloring questions on the +individual treads. \end{abstract} \maketitle @@ -190,6 +195,277 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. \end{remark} +\begin{proposition}[Source-side simple-cycle property] +\label{prop:no-level-d-pinch} +Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and +single-vertex source $v_0$. Let $d \geq 1$, $v \in L_d$, and let +$C'$ be a connected component of $G'_d$ such that $v$ is incident to +some face in $F_{C'}$. Then the depth-$d$ faces in $F_{C'}$ incident +to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$. + +Equivalently: for any such component, the source-side boundary of +$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$). +\end{proposition} + +\begin{proof} +Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$ +lie in two or more disjoint arcs of $v$'s rotation. Adjacent vertices +in $G$ differ in level by at most $1$, so a face at $v$ has depth +exactly $d$ iff both other vertices have level $\geq d$, and depth +$\leq d-1$ iff at least one has level $d-1$. Hence the gaps between +the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of +$v$, occurring in at least two disjoint arcs of $v$'s rotation. Pick +$p$ in one such gap and $q$ in another. + +The BFS ball $G[L_{d$ sub-regions, the inner outerplanar graph +$O$ captures the multi-hole structure as a disconnected or +non-$2$-connected outerplanar graph (with bridges or multiple +components), and its outer-face boundary closed walk serves as +$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices +multiple times. + +In the special case $d = 0$ with single-vertex source $S = \{v_0\}$, +$R_{C'}$ is the star of $v_0$, a topological closed disk with one +boundary cycle (the link of $v_0$); the corresponding tire graph has +degenerate outer boundary $\{v_0\}$. +\end{remark} + \begin{thebibliography}{9}