451 lines
20 KiB
TeX
451 lines
20 KiB
TeX
%% filename: amsart-template.tex
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%% American Mathematical Society
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%% ====================================================================
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\documentclass{amsart}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usetikzlibrary{backgrounds}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{conjecture}[theorem]{Conjecture}
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\numberwithin{equation}{section}
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\begin{document}
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\title{Medial Tire Cuts}
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% author one information
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\author{Eric Bauerfeld}
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\address{}
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\curraddr{}
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\email{}
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\thanks{}
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\subjclass[2010]{Primary }
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\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}
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\date{}
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\dedicatory{}
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\begin{abstract}
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Starting from the medial tire decomposition of a plane triangulation, we
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study the cuts that medial tires make in the full medial graph. We will
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show how to use medial tires to decompose the medial graph into a tree of
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three faces.
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\end{abstract}
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\maketitle
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\section{Introduction}
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This paper builds on the medial tire decomposition
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of~\cite{bauerfeld-medial-tire}. For a plane triangulation $G$ with
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fixed embedding we use freely the terminology and notation introduced
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there: the full medial graph $M(G)$, its decomposition into full medial
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tire graphs $\mathsf{M}(T)$ indexed by the treads $T$ of the tire tree
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$\mathcal{T}(G,S)$ at a level source $S$, the annular medial cycle
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$A(T)$, and the boundary medial vertex sets.
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We will show how to use medial tires to decompose the medial graph into
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a tree of three faces.
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\section{Cutting a full medial tire graph}
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We first describe a procedure that simultaneously \emph{labels} and
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\emph{cuts} a single full medial tire graph $\mathsf{M}(T)$ so that,
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after the cuts, the only faces are the outer face and $3$-faces
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(triangles)---the teeth of~\cite{bauerfeld-medial-tire}. The labelling
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assigns to each tooth an integer \emph{walk depth}; the cuts break the
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cyclic adjacencies of the teeth so that what remains is a tree of
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$3$-faces.
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By a \emph{cut} we mean the duplication of a single vertex of
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$\mathsf{M}(T)$: the vertex is split into two copies and the embedding is
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slit open along it (a planar unzip), separating the faces that meet only
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at that vertex. A cut therefore reduces the number of bounded faces that
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are not teeth.
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Throughout we use the teeth, up and down teeth, apexes, bites, the
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annular medial cycle $A(T)$, and the auxiliary plane graph $B(T)$
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of~\cite{bauerfeld-medial-tire}. Each tooth is a $3$-face of
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$\mathsf{M}(T)$, and the inner faces of $B(T)$ (the root face and the
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bite inner-gap faces) are the larger faces to be cut into teeth.
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\begin{definition}[Walk-depth labelling and cut]
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\label{def:walk-depth-cut}
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Let $\mathsf{M}(T)$ be a full medial tire graph. Assign walk depths and
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cuts as follows.
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\begin{enumerate}
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\item Pick an arbitrary up tooth, the \emph{entry tooth}. It has walk
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depth $d$.
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\item Traverse all the teeth that bound the inner face incident to the
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entry tooth clockwise until we reach the entry tooth, incrementing the
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walk depth by $1$ for each tooth traversed. (The \emph{inner face
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incident to the entry tooth} is the inner face of $B(T)$ whose boundary
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contains the annular edge of $A(T)$ carrying the entry tooth.)
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\item When you reach the last tooth in the face, perform a \emph{cut}
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by duplicating the annular vertex at which the traversal closes---the
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annular vertex of $A(T)$ shared by the last tooth and the entry tooth.
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\item Find the tooth $t$ with the highest walk depth which is a member
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of a bite.
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\item If $t$ is incident to a face $F$ with unlabelled teeth, traverse
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the teeth in $F$ starting from $t$ in the direction of the tooth
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incident to $t$ which is unlabelled, and increment the walk depth by
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$1$ as you travel. (Here a tooth is \emph{incident to $t$} when it
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shares an annular vertex of $A(T)$ with $t$.)
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\item Repeat steps (3)--(5) until all teeth have been labelled.
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\item Finally, perform an apex cut at every up tooth except an entry
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tooth. If the same medial vertex is the apex of two up teeth, do not
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cut that shared apex vertex.
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\end{enumerate}
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\end{definition}
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\begin{remark}[Entry and shared up-apex exceptions]
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\label{rem:up-apex-cut-exceptions}
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For a single full medial tire graph there is one entry tooth. In a
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chained tire decomposition each tread has its own entry tooth, inherited
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from the parent side or chosen at the root. These entry triangles are
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left uncut. Shared up-apex vertices are also left uncut: the intended
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cut set contains the apexes of singleton up teeth only.
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\end{remark}
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\begin{remark}[Closing tooth of a descended face]
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\label{rem:closing-tooth}
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For the entry face the traversal of step (2) closes by returning to the
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entry tooth, so the cut of step (3) duplicates the annular vertex shared
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by the last tooth and the entry tooth. For a face $F$ entered in step
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(5), the traversal instead closes upon reaching an already-labelled
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tooth: the other tooth of the bite through which $F$ was entered. In
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both cases the cut of step (3) duplicates the annular vertex shared by
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the last newly labelled tooth and this \emph{closing tooth}. Since both
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teeth of a bite are labelled while traversing its parent face, every
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descended face closes on such a tooth.
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\end{remark}
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\begin{example}[A worked walk-depth labelling and cut]
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\label{ex:worked-cut}
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Figure~\ref{fig:worked-cut} shows a full medial tire graph with annular
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cycle of length $8$, generated by the full medial tire generator
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of~\cite{bauerfeld-medial-tire}. Its eight teeth are: three up teeth on
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the annular edges $5,6,7$ in the root face; one bite pairing the annular
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edges $0$ and $4$; and three singleton down teeth on the annular edges
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$1,2,3$ lying in that bite's inner-gap face.
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Take the up tooth on edge $5$ as the entry tooth, with walk depth $0$.
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Its inner face is the root face, bounded by the teeth on edges
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$5,6,7,0,4$ in clockwise order. Step (2) labels them
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\[
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5\mapsto 0,\quad 6\mapsto 1,\quad 7\mapsto 2,\quad
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0\mapsto 3,\quad 4\mapsto 4,
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\]
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and step (3) cuts by duplicating the annular vertex $a_5$ shared by the
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last tooth (edge $4$) and the entry tooth (edge $5$). The highest-depth
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bite tooth is now the one on edge $4$ (walk depth $4$); it is incident to
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the still-unlabelled inner-gap face of the bite $(0,4)$. Entering that
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face from edge $4$ toward its unlabelled neighbour, step (5) labels the
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three down teeth
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\[
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3\mapsto 5,\quad 2\mapsto 6,\quad 1\mapsto 7,
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\]
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and closes on the already-labelled bite tooth of edge $0$, so step (3)
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cuts by duplicating the annular vertex $a_1$
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(Remark~\ref{rem:closing-tooth}). All eight teeth are now labelled, and
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the closing cuts are followed by apex cuts at the non-entry up teeth on
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edges $6$ and $7$. The labelling and cuts are produced by the script
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\texttt{lib/medial\_tire\_cut\_labelling.py}.
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\end{example}
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\begin{figure}[h]
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\centering
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\begin{tikzpicture}[scale=1.6,
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ann/.style={circle, fill=black, inner sep=1.0pt},
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upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
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downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
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bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
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cyc/.style={black, line width=1.0pt},
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tth/.style={black!55, line width=0.4pt},
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lbl/.style={font=\scriptsize},
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dlbl/.style={font=\scriptsize\bfseries, text=black},
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cut/.style={red!80!black, line width=1.3pt},
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cutlbl/.style={font=\tiny, text=red!75!black}]
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\draw[cyc] (0.000,1.000)--(0.707,0.707)--(1.000,0.000)--(0.707,-0.707)--(0.000,-1.000)--(-0.707,-0.707)--(-1.000,-0.000)--(-0.707,0.707)--cycle;
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\draw[tth] (-1.349,-0.559)--(-0.707,-0.707) (-1.349,-0.559)--(-1.000,-0.000);
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\draw[tth] (-1.349,0.559)--(-1.000,-0.000) (-1.349,0.559)--(-0.707,0.707);
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\draw[tth] (-0.559,1.349)--(-0.707,0.707) (-0.559,1.349)--(0.000,1.000);
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\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
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\draw[tth] (0.554,-0.230)--(1.000,0.000) (0.554,-0.230)--(0.707,-0.707);
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\draw[tth] (0.230,-0.554)--(0.707,-0.707) (0.230,-0.554)--(0.000,-1.000);
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\draw[tth] (0.000,-0.000)--(0.000,1.000) (0.000,-0.000)--(0.707,0.707);
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\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
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\node[ann] at (0.000,1.000) {};
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\node[ann] at (0.707,0.707) {};
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\node[ann] at (1.000,0.000) {};
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\node[ann] at (0.707,-0.707) {};
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\node[ann] at (0.000,-1.000) {};
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\node[ann] at (-0.707,-0.707) {};
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\node[ann] at (-1.000,-0.000) {};
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\node[ann] at (-0.707,0.707) {};
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\node[upv] at (-1.349,-0.559) {};
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\node[upv] at (-1.349,0.559) {};
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\node[upv] at (-0.559,1.349) {};
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\node[downv] at (0.554,0.230) {};
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\node[downv] at (0.554,-0.230) {};
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\node[downv] at (0.230,-0.554) {};
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\node[bitev] at (0.000,-0.000) {};
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\end{tikzpicture}
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\qquad
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\begin{tikzpicture}[scale=1.6,
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ann/.style={circle, fill=black, inner sep=1.0pt},
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upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
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downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
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bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
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cyc/.style={black, line width=1.0pt},
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tth/.style={black!55, line width=0.4pt},
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lbl/.style={font=\scriptsize},
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dlbl/.style={font=\scriptsize\bfseries, text=black},
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cut/.style={red!80!black, line width=1.3pt},
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cutlbl/.style={font=\tiny, text=red!75!black}]
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\draw[cyc] (0.000,1.000)--(0.707,0.707)--(1.000,0.000)--(0.707,-0.707)--(0.000,-1.000)--(-0.707,-0.707)--(-1.000,-0.000)--(-0.707,0.707)--cycle;
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\draw[tth] (-1.349,-0.559)--(-0.707,-0.707) (-1.349,-0.559)--(-1.000,-0.000);
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\draw[tth] (-1.349,0.559)--(-1.000,-0.000) (-1.349,0.559)--(-0.707,0.707);
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\draw[tth] (-0.559,1.349)--(-0.707,0.707) (-0.559,1.349)--(0.000,1.000);
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\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
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\draw[tth] (0.554,-0.230)--(1.000,0.000) (0.554,-0.230)--(0.707,-0.707);
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\draw[tth] (0.230,-0.554)--(0.707,-0.707) (0.230,-0.554)--(0.000,-1.000);
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\draw[tth] (0.000,-0.000)--(0.000,1.000) (0.000,-0.000)--(0.707,0.707);
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\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
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\node[ann] at (0.000,1.000) {};
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\node[ann] at (0.707,0.707) {};
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\node[ann] at (1.000,0.000) {};
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\node[ann] at (0.707,-0.707) {};
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\node[ann] at (0.000,-1.000) {};
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\node[ann] at (-0.707,-0.707) {};
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\node[ann] at (-1.000,-0.000) {};
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\node[ann] at (-0.707,0.707) {};
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\node[upv] at (-1.349,-0.559) {};
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\node[upv] at (-1.349,0.559) {};
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\node[upv] at (-0.559,1.349) {};
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\node[downv] at (0.554,0.230) {};
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\node[downv] at (0.554,-0.230) {};
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\node[downv] at (0.230,-0.554) {};
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\node[bitev] at (0.000,-0.000) {};
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\node[dlbl] at (0.177,0.427) {3};
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\node[dlbl] at (0.704,0.292) {7};
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\node[dlbl] at (0.704,-0.292) {6};
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\node[dlbl] at (0.292,-0.704) {5};
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\node[dlbl] at (-0.177,-0.427) {4};
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\node[dlbl] at (-1.101,-0.456) {0};
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\node[dlbl] at (-1.101,0.456) {1};
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\node[dlbl] at (-0.456,1.101) {2};
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\draw[cut] (-0.594,-0.594)--(-0.820,-0.820);
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\node[cutlbl] at (-0.919,-0.919) {cut 1};
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\draw[cut] (0.594,0.594)--(0.820,0.820);
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\node[cutlbl] at (0.919,0.919) {cut 2};
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\node[lbl, text=blue!60!black] at (-1.663,-0.689) {entry};
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\end{tikzpicture}
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\caption{A full medial tire graph (left) and its walk-depth labelling and
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cut (right), from Example~\ref{ex:worked-cut}. Black vertices are the
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annular medial vertices of the cycle $A(T)$; blue vertices are up-tooth
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apexes, red vertices are down-tooth apexes, and the larger red vertex is
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the shared apex of the bite on annular edges $0$ and $4$. On the right,
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each tooth carries its walk depth, and the two red slits mark the cuts:
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\emph{cut~1} duplicates $a_5$ as the root-face traversal closes, and
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\emph{cut~2} duplicates $a_1$ as the bite's inner-gap face closes. After
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the cuts the only bounded faces are the eight teeth.}
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\label{fig:worked-cut}
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\end{figure}
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\section{Chaining across the tire tree}
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Definition~\ref{def:walk-depth-cut} labels and cuts a single full medial
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tire graph. We extend it to the whole medial graph $M(G)$ through the
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medial tire decomposition of~\cite{bauerfeld-medial-tire}: the tire tree
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decomposes $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one per
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tread $T$, glued along their boundary medial vertices. A parent tread's
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inner level cycle is a child tread's outer level cycle, and the boundary
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medial vertices on that shared cycle belong to both treads.
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The key incidence is this. A \emph{boundary} (singleton) down tooth of a
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parent tread and the up tooth of the child tread glued to it across the
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shared level cycle have the \emph{same apex}: both apexes are the same
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medial vertex of $M(G)$, namely the medial vertex of an edge with both
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endpoints on the shared level cycle. We use this to carry the walk depth
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from a parent into its children.
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We label tread by tread, outward from the root:
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\begin{itemize}
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\item a tread with no parent in the decomposition---in particular the
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innermost recognised tread---is treated as a \emph{root} and entered at
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an arbitrary up tooth with walk depth $0$;
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\item a child tread is entered at the up tooth whose apex is the parent's
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boundary down tooth of lowest walk depth; that entry up tooth's walk depth
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is one more than that down tooth's, and the walk then increments locally
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within the child as in Definition~\ref{def:walk-depth-cut}.
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\end{itemize}
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The source cap contributes one additional cut before the recognised
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treads are assembled. If the root tread enters at an up tooth whose apex
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is the cap down tooth $xy$, we cut the cap annular vertex corresponding
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to the counter-clockwise source edge incident to $xy$. In the example of
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Figure~\ref{fig:whole-medial}, the root entry apex is the cap down tooth
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$14\!-\!4$, so the cap cut is placed at the medial vertex $14\!-\!5$.
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\begin{remark}[Candidate down teeth for chaining]
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\label{rem:chaining-candidates}
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The down teeth eligible to fix a child's entry are exactly the
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\emph{boundary} (singleton) down teeth of the parent: those lying in a
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single tread face, whose apex is the shared boundary medial vertex glued to
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a child up tooth. A bite's two down teeth are \emph{not} eligible. By the
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definition of a bite in~\cite{bauerfeld-medial-tire} its annular edge borders
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two tread faces, so a bite tooth is interior to the parent tread and its
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apex is a boundary medial vertex of no child. Hence ``the down tooth of
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lowest walk depth'' is read among the boundary down teeth only; a bite of
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even lower walk depth is skipped.
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\end{remark}
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Applying every tread's cuts to $M(G)$ assembles the per-tread labellings
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and cuts into a single cut graph of $M(G)$ together with a global
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walk-depth label map. This pipeline---random maximal planar graph, medial
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graph, tire decomposition at a vertex level source, and chained walk-depth
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labelling and cut---is carried out by the experiment script
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\texttt{experiments/run\_medial\_tire\_cut\_experiment.py}.
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\begin{example}[A medial tire cut from a random graph]
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\label{ex:real-cut}
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Run on a random maximal planar graph on $20$ vertices (seed $72$, level
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source vertex $9$), the experiment yields a single recognised tread
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$T_2$, drawn in Figure~\ref{fig:real-cut} with the walk-depth labelling
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and cut emitted by the graphics companion
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\texttt{experiments/draw\_medial\_tire\_cut.py}. Its annular cycle has
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length $8$, with up teeth on annular edges $0,3,4$, singleton down teeth
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on $1,6,7$, and a bite on the non-incident annular edges $2$ and $5$ (the
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central shared apex). Entering at the up tooth on edge $0$ with walk
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depth $0$, the root face is labelled in order ($0,1,2$ then $3,4,5$) and
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\emph{cut~1} duplicates $a_0$ as it closes; the walk then descends through
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the bite into its inner-gap face, labelling the two teeth there ($6,7$),
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and \emph{cut~2} duplicates $a_3$ as that face closes. The two cuts leave
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only the outer face and the eight teeth as $3$-faces.
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\end{example}
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\begin{figure}[h]
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|
\centering
|
|
\begin{tikzpicture}[scale=1.6,
|
|
ann/.style={circle, fill=black, inner sep=1.0pt},
|
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upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
|
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downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
|
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bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
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cyc/.style={black, line width=1.0pt},
|
|
tth/.style={black!55, line width=0.4pt},
|
|
lbl/.style={font=\scriptsize},
|
|
dlbl/.style={font=\scriptsize\bfseries, text=black},
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cut/.style={red!80!black, line width=1.3pt},
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cutlbl/.style={font=\tiny, text=red!75!black}]
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\draw[cyc] (0.000,1.000)--(0.707,0.707)--(1.000,0.000)--(0.707,-0.707)--(0.000,-1.000)--(-0.707,-0.707)--(-1.000,-0.000)--(-0.707,0.707)--cycle;
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|
\draw[tth] (0.559,1.349)--(0.000,1.000) (0.559,1.349)--(0.707,0.707);
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|
\draw[tth] (0.559,-1.349)--(0.707,-0.707) (0.559,-1.349)--(0.000,-1.000);
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|
\draw[tth] (-0.559,-1.349)--(0.000,-1.000) (-0.559,-1.349)--(-0.707,-0.707);
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|
\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
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|
\draw[tth] (-0.554,0.230)--(-1.000,-0.000) (-0.554,0.230)--(-0.707,0.707);
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|
\draw[tth] (-0.230,0.554)--(-0.707,0.707) (-0.230,0.554)--(0.000,1.000);
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|
\draw[tth] (0.000,-0.318)--(1.000,0.000) (0.000,-0.318)--(0.707,-0.707);
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|
\draw[tth] (0.000,-0.318)--(-0.707,-0.707) (0.000,-0.318)--(-1.000,-0.000);
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|
\node[ann] at (0.000,1.000) {};
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|
\node[ann] at (0.707,0.707) {};
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|
\node[ann] at (1.000,0.000) {};
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|
\node[ann] at (0.707,-0.707) {};
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|
\node[ann] at (0.000,-1.000) {};
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|
\node[ann] at (-0.707,-0.707) {};
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|
\node[ann] at (-1.000,-0.000) {};
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|
\node[ann] at (-0.707,0.707) {};
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|
\node[upv] at (0.559,1.349) {};
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|
\node[upv] at (0.559,-1.349) {};
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|
\node[upv] at (-0.559,-1.349) {};
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|
\node[downv] at (0.554,0.230) {};
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|
\node[downv] at (-0.554,0.230) {};
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|
\node[downv] at (-0.230,0.554) {};
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|
\node[bitev] at (0.000,-0.318) {};
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|
\node[dlbl] at (0.456,1.101) {0};
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|
\node[dlbl] at (0.704,0.292) {1};
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|
\node[dlbl] at (0.427,-0.336) {2};
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|
\node[dlbl] at (0.456,-1.101) {7};
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|
\node[dlbl] at (-0.456,-1.101) {6};
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|
\node[dlbl] at (-0.427,-0.336) {3};
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|
\node[dlbl] at (-0.704,0.292) {4};
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|
\node[dlbl] at (-0.292,0.704) {5};
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|
\draw[cut] (0.000,0.840)--(0.000,1.160);
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|
\node[cutlbl] at (0.000,1.300) {cut 1};
|
|
\draw[cut] (0.594,-0.594)--(0.820,-0.820);
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|
\node[cutlbl] at (0.919,-0.919) {cut 2};
|
|
\node[lbl, text=blue!60!black] at (0.689,1.663) {entry};
|
|
\end{tikzpicture}
|
|
\caption{The recognised tread $T_2$ of the medial tire decomposition of a
|
|
random maximal planar graph on $20$ vertices
|
|
(Example~\ref{ex:real-cut}), with its walk-depth labelling and cut. Black
|
|
vertices are the annular medial vertices of $A(T)$; blue vertices are
|
|
up-tooth apexes and red vertices down-tooth apexes, the larger red vertex
|
|
being the shared apex of the bite on annular edges $2$ and $5$. Each
|
|
tooth carries its walk depth; the red slits are the two cuts.}
|
|
\label{fig:real-cut}
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|
\end{figure}
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|
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|
Figure~\ref{fig:whole-medial} repeats the whole-medial-graph drawing on a
|
|
random maximal planar graph on $20$ vertices with minimum degree $5$
|
|
(plantri seed $59$, level source vertex $5$). The experiment recognises
|
|
two full medial tire treads, $T_1$ and $T_2$, and produces seven cuts:
|
|
one source-cap cut and six full-tread cuts. The
|
|
top panel shows the source triangulation with its level source
|
|
highlighted; the bottom panel draws $M(G)$ on the same straight-line
|
|
embedding by placing each medial vertex at the midpoint of its
|
|
corresponding source edge. Every medial vertex is labelled by that source
|
|
edge. Black vertices correspond to source edges joining consecutive
|
|
levels, and coloured vertices correspond to source edges within one level.
|
|
The red-highlighted vertices, walk-depth labels, and red slits are the
|
|
computed full-medial-tire labelling and cuts.
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\input{whole_medial_seed59_min5.tikz}
|
|
\caption{The source graph $G$ and the whole medial graph $M(G)$ of the
|
|
minimum-degree-$5$ maximal planar graph on $20$ vertices generated by
|
|
\texttt{plantri -m5} at seed $59$. The source vertex $5$ is highlighted
|
|
in the top panel. In the bottom panel, each medial vertex is placed at
|
|
the midpoint of its corresponding source edge and labelled by that edge.
|
|
Black vertices come from source edges between consecutive levels; coloured
|
|
vertices come from source edges within a single level of the chain. The
|
|
red-highlighted vertices, walk-depth labels, and seven red slits are the
|
|
computed source-cap cut and full-medial-tire labelling cuts for the
|
|
recognised treads $T_1$ and $T_2$. Drawn by
|
|
\texttt{experiments/draw\_medial\_tire\_cut.py} with
|
|
\texttt{--whole --min-degree 5}.}
|
|
\label{fig:whole-medial}
|
|
\end{figure}
|
|
|
|
\begin{thebibliography}{9}
|
|
|
|
\bibitem{bauerfeld-medial-tire}
|
|
E.~Bauerfeld,
|
|
\emph{Medial Tire Decompositions of Plane Triangulations},
|
|
manuscript (math-research repository), 2026.
|
|
|
|
\end{thebibliography}
|
|
|
|
\end{document}
|