From 47a87a4c2b49598e62bc1befcabaf42f14177d46 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 15:24:08 -0400 Subject: [PATCH] plane_depth_sequencing: generalise Lemma 2.6 to arbitrary outer-face source; tighten the citation in coloring_nested_tire_graphs MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit In `plane_depth_sequencing/paper.tex`: - Lemma 2.6 now allows any nonempty source S ⊆ V(G) whose vertices all lie on the boundary of the outer face of the chosen embedding, rather than only the outer-cycle case S = V(C). - The proof is the same argument with S in place of C: at d=0 each S-vertex remains on the outer face after restriction; for d ≥ 1 the BFS ball V_{ --- papers/coloring_nested_tire_graphs/paper.log | 6 +- papers/coloring_nested_tire_graphs/paper.pdf | Bin 464467 -> 464573 bytes papers/coloring_nested_tire_graphs/paper.tex | 14 +- papers/plane_depth_sequencing/paper.aux | 16 ++ papers/plane_depth_sequencing/paper.log | 251 +++++++++++++++++++ papers/plane_depth_sequencing/paper.pdf | Bin 240307 -> 250125 bytes papers/plane_depth_sequencing/paper.tex | 45 +++- 7 files changed, 316 insertions(+), 16 deletions(-) create mode 100644 papers/plane_depth_sequencing/paper.aux create mode 100644 papers/plane_depth_sequencing/paper.log diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index b6c5179..6625f02 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) 25 MAY 2026 15:17 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:23 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -214,7 +214,7 @@ LaTeX Warning: `h' float specifier changed to `ht'. 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zm>l^3T`@#rQ?zul_OK!6;OEcSTZLl)kxb1n4Y?AJ-8`Ll%{*~B@U#+)viC*4@LRyc zm#PZBK)*xY%E*HMi$bc~hEi9jyi-g`s=A;9Lt}R6{++KesnE{Vkb>kPS~3tzg&LQf z94e{elXjdqi<+N93il6Y7R8-K@KiD|+C^ z36qN6K|PvlaKNS;@QU9^{^yD&)0qC+P2gpPHc3nBLGkJH1Rvqf&jS6ix~u1n*&Zzz zXu1#+O1iJ}E?!k6OD79=4>wCw$NyE%W_C#K9;R*{Kin*>khpj_`2oC0G&E8w(n$XU DdW(A` diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index f92ab08..f5497b3 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -195,12 +195,14 @@ $F_{C'}$. \end{lemma} \begin{proof}[Proof sketch] -By Lemma~2.6 of \cite{bauerfeld-pds} (whose argument, given for an -outer-cycle source, extends verbatim to an arbitrary level source $S$ -by treating $S$ as the depth-$0$ set), the subgraph $G[L_{d'}]$ is -outerplanar for each $d' \geq 0$. Since subgraphs of outerplanar -graphs are outerplanar, both $G[V_{C'} \cap L_d]$ and -$G[V_{C'} \cap L_{d+1}]$ are outerplanar. +Since $S$ is a single vertex, we may choose a plane embedding of $G$ +that places $S$ on the outer face. By Lemma~2.6 of +\cite{bauerfeld-pds}, applied with this embedding and source set $S$, +the subgraph $G[L_{d'}]$ is outerplanar for each $d' \geq 0$; +outerplanarity is a graph property, so this conclusion is independent +of the embedding choice. Since subgraphs of outerplanar graphs are +outerplanar, both $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are +outerplanar. Layer containment (a consequence of the bounded-step property of BFS on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$ diff --git a/papers/plane_depth_sequencing/paper.aux b/papers/plane_depth_sequencing/paper.aux new file mode 100644 index 0000000..dda1255 --- /dev/null +++ b/papers/plane_depth_sequencing/paper.aux @@ -0,0 +1,16 @@ +\relax +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent } +\citation{baker1994} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{2}{}\protected@file@percent } +\newlabel{lem:outerplanarity}{{2.6}{2}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Quadrilateral sequencing}}{3}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The deep embedding $G'$ of a small maximal planar graph (drawn with one outer-cap face as the outer face), with each quadrilateral $Q_n$ of the canonical sequence labelled by its index and the move code (AD = anchor drop, LA = level add, J = join, RC = ring completion) of the move that produced it. Solid edges are non-level; dashed edges are level. Background colour encodes quadrilateral type: amber for shallow diamonds, teal for deep diamonds, pink for S quads. Outer-cycle vertices are blue, the outer-cap vertex $x^{*}$ is red. 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b/papers/plane_depth_sequencing/paper.tex index b35fc4c..41e558a 100644 --- a/papers/plane_depth_sequencing/paper.tex +++ b/papers/plane_depth_sequencing/paper.tex @@ -107,7 +107,7 @@ A triangle $\{u, v, w\}$ in $G$ is an \emph{up triangle} if the multiset of dept \end{definition} \begin{remark} -We now relate our terminology to existing terminology, namely $k$-outerplanar graphs \cite{baker1994}. The following definition and lemma show that the subgraph induced by any single depth level is outerplanar, i.e., $1$-outerplanar in the sense of Baker. +We now relate our terminology to existing terminology, namely $k$-outerplanar graphs \cite{baker1994}. The following definition and lemma show that the subgraph induced by any single depth level relative to any source set on the outer face is outerplanar, i.e., $1$-outerplanar in the sense of Baker. \end{remark} \begin{definition} @@ -115,19 +115,50 @@ A plane graph is \emph{outerplanar} if every vertex lies on the outer face. More \end{definition} \begin{lemma} -Let $G$ be a graph with a plane embedding and outer cycle $C$. For each $d \geq 0$, the subgraph of $G$ induced by $V_d = \{v \in V(G) : \mathrm{depth}(v) = d\}$ is outerplanar. +\label{lem:outerplanarity} +Let $G$ be a planar graph with a plane embedding $\Pi$, and let +$S \subseteq V(G)$ be a nonempty set of vertices, every one of which +lies on the boundary of the outer face of $\Pi$. For each $d \geq 0$, +the subgraph of $G$ induced by +\[ + V_d^S := \{ v \in V(G) : \mathrm{dist}_G(v, S) = d \} +\] +is outerplanar. + +The special case $S = V(C)$, where $C$ is the outer cycle, recovers +$V_d^S = V_d$ (depth-$d$ vertices as in Definition~2.1) and is the +form most often used in the rest of this paper. \end{lemma} \begin{proof} -Let $H = G[V_d]$ with the plane embedding inherited from $G$. It suffices to show every vertex of $H$ lies on the outer face of $H$. +Let $H = G[V_d^S]$ with the plane embedding inherited from $\Pi$. It +suffices to show that every vertex of $H$ lies on the outer face of $H$. -For $d = 0$, we have $V_0 = C$, so $H$ is outerplanar. +For $d = 0$, $V_0^S = S$, and by hypothesis every vertex of $S$ lies +on the boundary of the outer face of $\Pi$. Removing the vertices and +edges of $G \setminus H$ from the embedding only enlarges or merges +face regions, so the outer face of $\Pi$ is contained in the outer face +of $H$, and every vertex of $S$ remains on the outer face of $H$. -For $d \geq 1$, let $U$ be the open subset of the plane obtained by removing all vertices and edges of $H$. We show every $v \in V_d$ lies on the boundary of the component $U_{\mathrm{out}}$ of $U$ containing the outer face of $G$. +For $d \geq 1$, let $U$ be the open subset of the plane obtained by +removing all vertices and edges of $H$. We show every $v \in V_d^S$ +lies on the boundary of the component $U_{\mathrm{out}}$ of $U$ +containing the outer face of $\Pi$. -Since every vertex in $V_{\leq d-1}$ has a shortest path to $C$ passing entirely through $V_{\leq d-1}$, the subgraph $G[V_{\leq d-1}]$ is connected and contains $C$. Its vertices and edges lie in $U$ (as they are not in $H$), and $C$ borders the outer face of $G$, so $G[V_{\leq d-1}]$ and the outer face of $G$ are connected within $U$, hence both lie in $U_{\mathrm{out}}$. +Since every vertex in $V_{