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split: extract foundational depth material into new plane_depth paper

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Splits the existing plane_depth_sequencing paper into two:

  papers/plane_depth/paper.tex (NEW, 4 pages):
    - Plane depth definition.
    - Level edge, up/down/neutral triangle classification.
    - Outerplanarity lemma (formerly Lemma 2.6 of PDS).
    - Deep embedding G' definition.
    - "Every face of G' is up or down" lemma.
    - Unique level edge per face; shared level edge between adjacent faces.
    - Quadrilateral decomposition definition with three types
      (shallow diamond, deep diamond, S quad).

  papers/plane_depth_sequencing/paper.tex (slimmed from 11 → 6 pages):
    - Cites plane_depth for all foundational definitions.
    - Keeps: slice, move definitions (anchor drop, level add, join,
      ring completion), move selection, termination theorem.

  papers/coloring_nested_tire_graphs/paper.tex:
    - Bibliography updated: cite bauerfeld-depth instead of bauerfeld-pds.
    - Two in-text references updated to cite the new outerplanarity
      lemma in plane_depth.

Rationale: the outerplanarity / deep-embedding / quadrilateral-
decomposition material is foundational and reused by multiple
papers (and by the proposed level-cycle generalization).  The
quadrilateral-sequencing programme is one specific application.
Splitting lets coloring_nested_tire_graphs cite the foundations
cleanly without dragging in the sequencing machinery.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 13:49:44 -04:00
parent 1513922dec
commit 8b6c2b621c
12 changed files with 904 additions and 301 deletions
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\relax
\citation{bauerfeld-pds-seq}
\citation{bauerfeld-nested-tires}
\citation{baker1994}
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\documentclass{amsart}
\usepackage{amssymb}
\usepackage{graphicx}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
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\begin{document}
\title{Plane Depth}
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, plane depth, level edge, deep
embedding, quadrilateral decomposition, $k$-outerplanar graph}
\date{}
\dedicatory{}
\begin{abstract}
Given a plane embedding of a graph with outer cycle $C$, the
\emph{plane depth} of a vertex is its graph distance to $C$. We
develop this depth function into a layered combinatorial structure on
plane triangulations: the subgraph induced by each depth level is
outerplanar (recovering Baker's notion of a $k$-outerplanar graph);
each triangular face is classified by its depth multiset as
\emph{up}, \emph{down}, or \emph{neutral}; and the \emph{deep
embedding} of a maximal planar graph, obtained by inserting a vertex
into every neutral face (including the outer face), has every face
either up or down. Pairing adjacent triangles across their unique
level edge yields a \emph{quadrilateral decomposition} of the
spherical deep embedding into three combinatorial types: shallow
diamonds, deep diamonds, and S quads.
This paper isolates the foundational depth-and-decomposition material
that supports several downstream applications --- including the
quadrilateral sequencing of \cite{bauerfeld-pds-seq} and the
nested-tire colouring framework of \cite{bauerfeld-nested-tires}.
\end{abstract}
\maketitle
\section{Definitions}
\begin{definition}
\label{def:plane-depth}
Let $G$ be a graph with a plane embedding, and let $C$ be the outer
cycle of that embedding. The \emph{plane depth} of a vertex
$v \in V(G)$ relative to the embedding and $C$ is
\[
\mathrm{depth}(v) = \min_{u \in V(C)} d(v, u),
\]
where $d(v, u)$ denotes the graph distance between $v$ and $u$ in
$G$.
\end{definition}
\begin{definition}
\label{def:level-edge}
An edge $\{u, v\} \in E(G)$ is a \emph{level edge} if
$\mathrm{depth}(u) = \mathrm{depth}(v)$.
\end{definition}
\begin{definition}
\label{def:triangle-types}
A triangle $\{u, v, w\}$ in $G$ is an \emph{up triangle} if the
multiset of depths of its vertices is $\{d, d+1, d+1\}$ for some
$d \geq 0$, a \emph{down triangle} if the multiset of depths is
$\{d, d, d+1\}$ for some $d \geq 0$, and a \emph{neutral triangle} if
the multiset of depths is $\{d, d, d\}$ for some $d \geq 0$.
\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
relative to any source set on the outer face is outerplanar, i.e.\
$1$-outerplanar in the sense of Baker.
\end{remark}
\begin{definition}
A plane graph is \emph{outerplanar} if every vertex lies on the outer
face. More generally, a plane graph is \emph{$k$-outerplanar} for
$k \geq 1$ if removing all vertices on the outer face yields a
$(k-1)$-outerplanar graph, where every graph on the empty vertex set
is $0$-outerplanar.
\end{definition}
\section{Outerplanarity of depth levels}
\begin{lemma}
\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~\ref{def:plane-depth}) and is the form most often used in
applications.
\end{lemma}
\begin{proof}
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 $d = 0$, $V_0^S = S$, 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 $v \in V_d^S$
lies on the boundary of the component $U_{\mathrm{out}}$ of $U$
containing the outer face of $\Pi$.
Since every vertex in $V_{<d}^S := \bigcup_{e < d} V_e^S$ has a
shortest path to $S$ passing entirely through $V_{<d}^S$, the
subgraph $G[V_{<d}^S]$ is connected and contains $S$. Its vertices
and edges lie in $U$ (none belong to $H$), and $S$ borders the outer
face of $\Pi$, so $G[V_{<d}^S]$ and the outer face of $\Pi$ are
connected within $U$, hence both lie in $U_{\mathrm{out}}$.
Now let $v \in V_d^S$. Since $d \geq 1$, there exists
$u \in V_{d-1}^S$ adjacent to $v$ in $G$. The edge $\{v, u\}$ is not
in $H$, so it lies in $U$. Since $u \in V_{d-1}^S \subseteq
U_{\mathrm{out}}$ and $\{v, u\}$ is a connected subset of $U$
containing $u$, the entire edge lies in $U_{\mathrm{out}}$. The
vertex $v$ is an endpoint of this edge but is not in $U$, so $v$ lies
on the boundary of $U_{\mathrm{out}}$, i.e.\ on the outer face of
$H$.
\end{proof}
\section{Deep embedding}
\begin{definition}
\label{def:deep-embedding}
Let $G$ be a maximal planar graph with a plane embedding and outer
cycle $C$. The \emph{deep embedding} of $G$ is the graph $G'$
obtained from $G$ by the following operation: for every neutral
triangular face $\{u, v, w\}$ of $G$ --- \emph{including the outer
face}, whose vertices are the three vertices of $C$ --- add a new
vertex $x$ placed in that face and adjacent to each of $u$, $v$,
and $w$. The vertex added inside the outer face is denoted $x^*$
and called the \emph{outer-cap vertex}; the three triangular faces
it induces with the edges of $C$ are the \emph{outer-cap faces}.
We henceforth view $G'$ as embedded on the sphere $S^2$, with no
distinguished outer face.
\end{definition}
\begin{lemma}
\label{lem:up-down-faces}
Let $G'$ be the deep embedding of a maximal planar graph $G$. Every
face of $G'$ is either an up triangle or a down triangle.
\end{lemma}
\begin{proof}
We first establish that for any edge $\{p, q\}$ in $G$, the depths of
$p$ and $q$ differ by at most $1$. Suppose for contradiction that
$\mathrm{depth}(p) = d$ and $\mathrm{depth}(q) = d + n$ for some
$n \geq 2$. Since $\mathrm{depth}(p) = d$, there exists a path of
length $d$ from $p$ to some vertex of $C$. Prepending the edge
$\{q, p\}$ gives a path of length $d + 1$ from $q$ to $C$, so
$\mathrm{depth}(q) \leq d + 1 < d + n$, a contradiction. The case
$\mathrm{depth}(q) = d - n$ is handled identically: there exists a
path of length $d - n$ from $q$ to some vertex of $C$, and prepending
the edge $\{p, q\}$ gives a path of length $d - n + 1 \leq d - 1 < d$
from $p$ to $C$, contradicting $\mathrm{depth}(p) = d$.
Since $G$ is a triangulation, every interior face of $G$ is a
triangle $\{u, v, w\}$ with all three pairs adjacent. By the above,
each pair of vertices in a triangle differs in depth by at most $1$,
so no triangle can contain vertices of depths $d$ and $d + 2$
simultaneously. The possible depth patterns for a triangle in $G$
are therefore exactly a neutral triangle, a down triangle, or an up
triangle.
We now consider each case under the deep embedding.
\emph{Case 1: up triangle or down triangle.} These triangles are
not modified by the deep embedding, so they remain as faces of $G'$,
satisfying the lemma.
\emph{Case 2: neutral triangle.} The deep embedding inserts a new
vertex $x$ adjacent to $u$, $v$, and $w$, replacing the face
$\{u, v, w\}$ with three new faces $\{u, v, x\}$, $\{v, w, x\}$, and
$\{u, w, x\}$. It remains to determine the depth of $x$ in $G'$.
Since $x$ is adjacent only to $u$, $v$, and $w$, every path in $G'$
from $x$ to $C$ must pass through one of them, so $x$ has strictly
greater depth than $u$, $v$, and $w$. Each of the three new faces
is thus a down triangle, satisfying the lemma. The same argument
applies to the outer face: the outer-cap vertex $x^*$ is adjacent to
all three vertices of $C$ (which lie at depth $0$), so
$\mathrm{depth}(x^*) = 1$, and each of the three outer-cap faces is
a down triangle.
Since every face of $G'$ falls into one of these cases, the result
follows.
\end{proof}
\section{Quadrilateral decomposition}
\begin{lemma}
\label{lem:unique-level-edge}
Every interior face of $G'$ has exactly one level edge.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:up-down-faces}, each interior face is an up
triangle (depths $\{d, d+1, d+1\}$) or a down triangle (depths
$\{d, d, d+1\}$). In both cases, exactly one of the three vertex
pairs has equal depth.
\end{proof}
\begin{lemma}
\label{lem:shared-level-edge}
Let $e = \{p, q\}$ be any level edge of $G'$. Then $e$ is the
unique level edge of both faces incident to it.
\end{lemma}
\begin{proof}
On the sphere, both faces $T, T'$ incident to $e$ are triangles.
Since $p$ and $q$ have equal depth, $e$ is a level edge of $T$ and
of $T'$, and by Lemma~\ref{lem:unique-level-edge} each has $e$ as
its unique level edge.
\end{proof}
\begin{definition}
\label{def:quad-decomposition}
The \emph{quadrilateral decomposition} of $G'$ pairs each face of
$G'$ with the face on the other side of its (unique) level edge.
Each pair, together with the four non-level edges of the two
triangles, bounds a \emph{quadrilateral} of the decomposition.
\end{definition}
\begin{remark}
Because $G'$ is taken on the sphere, every edge lies between two
triangular faces, so the pairing above applies uniformly. In
particular, each edge of $C$ is a level edge shared between one
interior boundary down triangle (depths $\{0, 0, 1\}$, with the
depth-$1$ vertex inside $C$) and one outer-cap down triangle
(depths $\{0, 0, 1\}$, with apex $x^*$). The three resulting
quadrilaterals, one per edge of $C$, are the \emph{boundary deep
diamonds}; they are the outermost quadrilaterals of the
decomposition.
\end{remark}
\begin{definition}
\label{def:quad-types}
Each quadrilateral is one of three types, classified by the depths
of its two non-level vertices relative to the depth $d$ of the
shared level edge:
\begin{itemize}
\item a \emph{shallow diamond}, formed by two up triangles, with
vertex depths $(d-1, d, d-1, d)$ around the boundary;
\item a \emph{deep diamond}, formed by two down triangles, with
vertex depths $(d+1, d, d+1, d)$ around the boundary;
\item an \emph{S quad}, formed by one up and one down triangle,
with vertex depths $(d-1, d, d+1, d)$ around the boundary.
\end{itemize}
\end{definition}
\begin{thebibliography}{9}
\bibitem{baker1994}
B.~S.~Baker,
\emph{Approximation algorithms for {NP}-complete problems on planar
graphs},
Journal of the ACM, vol.~41, no.~1, pp.~153--180, 1994.
\bibitem{bauerfeld-pds-seq}
E.~Bauerfeld,
\emph{Plane Depth Sequencing},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-nested-tires}
E.~Bauerfeld,
\emph{Coloring Nested Tire Graphs},
manuscript (math-research repository), 2026.
\end{thebibliography}
\end{document}