coloring_nested_tire_graphs: firm up Lemma 1.7 with explicit manifold/boundary hypotheses
Adds two explicit hypotheses (R1) and (R2) to Lemma 1.7 (tire-component
lemma) and tightens the proof to use them precisely:
(R1) R_{C'} is a topological 2-manifold with boundary; equivalently,
at every v ∈ V_{C'} the depth-d faces of F_{C'} incident to v form
a single contiguous arc in v's rotation in Π_G.
(R2) R_{C'} has at most two boundary components.
These rule out, respectively, pinch points (where C' wraps around a
vertex via a global path of depth-d faces, producing a non-manifold
region) and multi-hole topology (a "pair of pants" or worse, which can
occur when several disjoint depth->d lobes sit inside one depth-d
component).
The proof is reorganised into labelled steps:
1. Outerplanarity of the two level parts (via Lemma 2.6 of [1]).
2. Layer containment V_{C'} ⊆ L_d ∪ L_{d+1}.
3. Boundary edges are monochromatic in level (full case analysis
on the third vertex of the outside face f', using the
bounded-step property of δ).
4. Boundary components are simple cycles (uses R1: locally at any
boundary point R_{C'} is a half-disk, so the boundary walk
visits each vertex once).
5. Topological type: planarity + R1 + R2 forces disk or annulus
via the classification of compact orientable 2-manifolds with
boundary.
6. Tire structure: identifies the two boundary parts as the level-d
and level-(d+1) induced subgraphs, in either order.
Adds Remark 1.10 documenting when (R1) and (R2) hold or fail:
- (R1) fails iff there is a pinch vertex whose cyclic level sequence
around it enters and leaves {d, d+1} more than once.
- (R2) fails iff R_{C'} encloses two or more disjoint depth->d
sub-regions (the depth-<d region is always a single connected
BFS ball, so the multi-hole obstruction is always on the away side).
- Notes the d=0 single-vertex-source case satisfies both hypotheses
automatically (R_{C'} is the star of v_0).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -9,12 +9,13 @@
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\@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 }
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\newlabel{fig:tire-example}{{2}{3}}
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\newlabel{lem:tire-component}{{1.7}{3}}
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\newlabel{rem:tire-component-degenerate}{{1.8}{4}}
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\bibcite{bauerfeld-pds}{1}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{12.7778pt}
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\newlabel{rem:tire-component-degenerate}{{1.8}{4}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
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\gdef \@abspage@last{4}
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\newlabel{rem:R1-R2-when}{{1.9}{5}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
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@@ -182,59 +182,110 @@ $S \subseteq V(G)$ be a level source. For $d \geq 0$, let
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be the inner-dual subgraph on dual vertices of dual depth $d$, and let
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$C'$ be a connected component of $G'_d$. Write
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$F_{C'} := \{f : d_f \in V(C')\}$,
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$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and
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$C := G[V_{C'}]$ with the embedding inherited from $\Pi_G$.
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$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and let $C := G[V_{C'}]$ inherit
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its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
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\subseteq |\Pi_G|$.
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Then $C$, together with its inherited embedding, is a tire graph in the
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sense of Definition~\ref{def:tire-graph}: the two boundary parts
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$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ of $C$ are the level-$d$
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subgraph $G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph
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Assume:
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\begin{itemize}
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\item[\emph{(R1)}] $R_{C'}$ is a topological $2$-manifold with boundary;
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equivalently, at every $v \in V_{C'}$ the faces of $F_{C'}$
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incident to $v$ form a single contiguous arc in the rotation
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around $v$ in $\Pi_G$.
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\item[\emph{(R2)}] $R_{C'}$ has at most two boundary components.
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\end{itemize}
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Then $C$, with the inherited embedding, is a tire graph in the sense of
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Definition~\ref{def:tire-graph}: its two boundary parts
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$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ are the level-$d$ subgraph
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$G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph
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$G[V_{C'} \cap L_{d+1}]$, in either order, and the triangular faces of
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$C$ inside its closed boundary region are exactly the faces of $G$ in
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$C$ inside the closed boundary region are exactly the faces of $G$ in
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$F_{C'}$.
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\end{lemma}
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\begin{proof}[Proof sketch]
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Since $S$ is a single vertex, we may choose a plane embedding of $G$
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that places $S$ on the outer face. By Lemma~2.6 of
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\cite{bauerfeld-pds}, applied with this embedding and source set $S$,
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the subgraph $G[L_{d'}]$ is outerplanar for each $d' \geq 0$;
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outerplanarity is a graph property, so this conclusion is independent
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of the embedding choice. Since subgraphs of outerplanar graphs are
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outerplanar, both $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are
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outerplanar.
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\begin{proof}
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\emph{Outerplanarity of the two level parts.} Since $S$ is a single
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vertex, choose a plane embedding of $G$ with $S$ on the outer face.
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By Lemma~2.6 of \cite{bauerfeld-pds} applied with this embedding and
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source $S$, $G[L_{d'}]$ is outerplanar for each $d' \geq 0$;
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outerplanarity is a graph property, so the conclusion is independent
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of the embedding choice. Subgraphs of outerplanar graphs are
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outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are
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both outerplanar.
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Layer containment (a consequence of the bounded-step property of BFS
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on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$
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has vertex partition $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap
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L_{d+1})$, and every face $f \in F_{C'}$ is a triangle with at least
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one vertex in $L_d$.
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\emph{Layer containment.} Each $f \in F_{C'}$ has at least one vertex
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at level $d$, and adjacent vertices in $G$ differ in level by at most
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$1$; combined with $\delta_G(d_f) = d$, this forces
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$V(f) \subseteq L_d \cup L_{d+1}$. Hence $V_{C'} \subseteq L_d \cup
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L_{d+1}$, and $C$ has vertex partition
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$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
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It remains to identify the boundary of $R_{C'} := \bigcup_{f \in
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F_{C'}} f \subseteq |\Pi_G|$. Each edge on the topological boundary
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$\partial R_{C'}$ separates a face $f \in F_{C'}$ (depth $d$) from a
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face $f' \notin F_{C'}$. Such an $f'$ has dual depth in $\{d-1, d+1\}$:
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if $d$, then $f'$ shares an edge with a depth-$d$ face but lies in a
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distinct component of $G'_d$, which contradicts the connectivity of
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$C'$ together with the fact that adjacent depth-$d$ dual vertices belong
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to the same component. A short case analysis on the level of the third
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vertex of $f'$ shows that boundary edges with $\delta(f') = d-1$ have
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both endpoints in $L_d$, while those with $\delta(f') = d+1$ have both
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endpoints in $L_{d+1}$. Each connected component of $\partial R_{C'}$
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is therefore monochromatic in level.
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\emph{Boundary edges are monochromatic in level.} Each edge $e$ on
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$\partial R_{C'}$ separates a face $f \in F_{C'}$ from a face
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$f' \notin F_{C'}$. Because $f$ and $f'$ share the edge $e$, their
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dual vertices are adjacent in $G'$; if both had depth $d$ they would
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lie in the same component of $G'_d$, contradicting $d_{f} \in C'$ and
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$d_{f'} \notin C'$. Hence $\delta_G(d_{f'}) \neq d$; combined with
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the bounded-step property of $\delta$ across $G'$-adjacent faces,
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$\delta_G(d_{f'}) \in \{d-1, d+1\}$.
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\begin{itemize}
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\item If $\delta_G(d_{f'}) = d - 1$, the third vertex $w$ of
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$f' = \{u, v, w\}$ (where $u, v$ are the endpoints of $e$) has
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$\ell(w) = d - 1$. Each of $u, v$ has $\ell \in \{d, d+1\}$
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(from $V(f) \subseteq L_d \cup L_{d+1}$) and is adjacent to $w$,
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forcing $\ell(u), \ell(v) \in \{d-2, d-1, d\} \cap \{d, d+1\} =
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\{d\}$.
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\item If $\delta_G(d_{f'}) = d + 1$, then all three vertices of $f'$
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lie in $L_{\geq d+1}$, so in particular $\ell(u) = \ell(v) =
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d + 1$.
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\end{itemize}
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Each connected boundary component thus carries a single type at every
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edge: any vertex on a boundary component has two boundary edges
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incident to it (by R1, see below), both of the same type, so its
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level is fixed. Therefore each boundary component of $\partial R_{C'}$
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is monochromatic in level.
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Since $C'$ is connected and $R_{C'}$ is a connected planar 2-complex
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of triangles glued along edges, $\partial R_{C'}$ consists of either
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one closed walk (when $R_{C'}$ is a topological disk) or two closed
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walks (when $R_{C'}$ is a topological annulus). These walks are
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simple cycles in $G$ on the respective level sets (with one cycle
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possibly degenerating to a single vertex of $L_d$ or $L_{d+1}$ at the
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endpoints of the BFS, giving the degenerate-boundary case of
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Definition~\ref{def:tire-graph}). Together with the outerplanarity of
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$G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ established above,
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these cycles serve as the two boundary parts of $C$, in either order,
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and the depth-$d$ triangles in $F_{C'}$ tile the closed region between
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them.
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\emph{Boundary components are simple cycles.} By hypothesis (R1),
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$R_{C'}$ is a $2$-manifold with boundary, so locally at any boundary
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point $p$ the region $R_{C'}$ is homeomorphic to a half-disk and the
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link of $p$ in $\partial R_{C'}$ is an arc with two endpoints. In
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particular, at every boundary vertex $v$ exactly two boundary edges
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are incident, and the boundary walk traverses $v$ exactly once. Each
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boundary component is therefore a simple closed walk in $G$ --- a
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simple cycle, possibly degenerating to a single vertex if $v$ has no
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incident boundary edges (which happens precisely at the BFS endpoints
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$d = 0$ with $S = \{v_0\}$, or where an entire level set $V_{C'} \cap
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L_{d+1}$ is empty).
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\emph{Topological type.} $R_{C'}$ is a connected, compact, planar
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$2$-manifold with boundary; planarity gives orientability and genus
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zero, so by the classification of surfaces $R_{C'}$ is homeomorphic
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to a closed disk with $n - 1$ open disks removed, where $n \geq 1$ is
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the number of boundary components. Hypothesis (R2) gives $n \leq 2$,
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so $R_{C'}$ is either a closed disk ($n = 1$) or a closed annulus
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($n = 2$).
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\emph{Tire structure.} In the annulus case ($n = 2$), the two
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boundary cycles are simple cycles on $L_d$ and $L_{d+1}$ respectively
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(by the previous two paragraphs). These are the cycles bounding the
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two outerplanar subgraphs $G[V_{C'} \cap L_d]$ and
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$G[V_{C'} \cap L_{d+1}]$ in $\Pi_G$, and they meet the
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tire-graph definition with $B_{\mathrm{out}} \in \{$ those cycles $\}$
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in either order. In the disk case ($n = 1$), the unique boundary
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cycle lies on one of the two levels, and the other level set of
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$V_{C'}$ is either empty or a single interior vertex of the disk
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(the BFS endpoint). When it is a single vertex this is the
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degenerate-boundary case of Definition~\ref{def:tire-graph}; the
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remaining case ($V_{C'} \cap L_{d+1} = \emptyset$, which arises at
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$d = D_{\max}$) is excluded by interpreting one boundary part as a
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degenerate single vertex on $L_{d+1}$ (taken empty by convention,
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which we omit here).
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The triangular faces inside the closed boundary region of $C$ are by
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construction the depth-$d$ faces in $F_{C'}$, and the edges of $C$ are
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$E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}$ where
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$E_{\mathrm{ann}}$ are the edges of $G$ between $V_{C'} \cap L_d$ and
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$V_{C'} \cap L_{d+1}$ that bound a face of $F_{C'}$.
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\end{proof}
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\begin{remark}
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@@ -249,6 +300,42 @@ the orientation of the inherited embedding (equivalently, on which side
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of $C$ contains the rest of $\Pi_G$).
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\end{remark}
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\begin{remark}
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\label{rem:R1-R2-when}
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The two hypotheses of Lemma~\ref{lem:tire-component} hold in many
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natural settings but can fail in general:
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\emph{(R1) and the pinch obstruction.} Hypothesis (R1) fails at a
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\emph{pinch vertex} $v \in V_{C'}$ when the faces of $F_{C'}$ incident
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to $v$ split into two or more disjoint arcs of the rotation around $v$
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in $\Pi_G$. Such a $v$ has at least four neighbours $w_i, w_{i+1},
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w_j, w_{j+1}$ (with $i + 1 < j$) in cyclic order such that the faces
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$\{v, w_i, w_{i+1}\}$ and $\{v, w_j, w_{j+1}\}$ are both depth-$d$
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(both endpoints at level $\geq d$) while at least one face in each of
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the rotation gaps between them carries depth $\neq d$. Concretely,
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this occurs precisely when the cyclic level sequence
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$\ell(w_1), \ldots, \ell(w_{\deg v})$ enters and leaves $\{d, d+1\}$
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more than once. Whenever such a $v$ exists and the two arcs are
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joined to a common component of $G'_d$ by some \emph{other} path of
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depth-$d$ faces (not through $v$), the resulting $R_{C'}$ is a wedge
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of two manifold regions at $v$, violating (R1).
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\emph{(R2) and the multi-hole obstruction.} Hypothesis (R2) fails
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when the depth-$d$ region $R_{C'}$ encloses two or more disjoint
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depth-$> d$ sub-regions. In a BFS the depth-$< d$ region (the BFS
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ball of radius $d - 1$) is connected, so at most one boundary
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component of $R_{C'}$ can lie on the source side; (R2) is therefore
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equivalent to ``the closure of the depth-$> d$ region adjacent to
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$R_{C'}$ has at most one connected component.'' Multi-hole topology
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arises when several disjoint depth-$> d$ ``lobes'' of $G$ sit inside
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the same depth-$d$ component.
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|
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In the special case $d = 0$ with single-vertex source $S = \{v_0\}$
|
||||
both hypotheses hold automatically: $R_{C'}$ is the star of $v_0$,
|
||||
a topological closed disk with one boundary cycle (the link of $v_0$),
|
||||
giving a tire graph with degenerate inner boundary $\{v_0\}$.
|
||||
\end{remark}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{bauerfeld-pds}
|
||||
|
||||
Reference in New Issue
Block a user