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\begin{document}

\title{Medial Tire Cuts}

%    author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}

\subjclass[2010]{Primary }

\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}

\date{}

\dedicatory{}

\begin{abstract}
Starting from the medial tire decomposition of a plane triangulation, we
study the cuts that medial tires make in the full medial graph.  We will
show how to use medial tires to decompose the medial graph into a tree of
three faces.
\end{abstract}

\maketitle

\section{Introduction}

This paper builds on the medial tire decomposition
of~\cite{bauerfeld-medial-tire}.  For a plane triangulation $G$ with
fixed embedding we use freely the terminology and notation introduced
there: the full medial graph $M(G)$, its decomposition into full medial
tire graphs $\mathsf{M}(T)$ indexed by the treads $T$ of the tire tree
$\mathcal{T}(G,S)$ at a level source $S$, the annular medial cycle
$A(T)$, and the boundary medial vertex sets.

We will show how to use medial tires to decompose the medial graph into
a tree of three faces.

\section{Cutting a full medial tire graph}

We first describe a procedure that simultaneously \emph{labels} and
\emph{cuts} a single full medial tire graph $\mathsf{M}(T)$ so that,
after the cuts, the only faces are the outer face and $3$-faces
(triangles)---the teeth of~\cite{bauerfeld-medial-tire}.  The labelling
assigns to each tooth an integer \emph{walk depth}; the cuts break the
cyclic adjacencies of the teeth so that what remains is a tree of
$3$-faces.

By a \emph{cut} we mean the duplication of a single vertex of
$\mathsf{M}(T)$: the vertex is split into two copies and the embedding is
slit open along it (a planar unzip), separating the faces that meet only
at that vertex.  A cut therefore reduces the number of bounded faces that
are not teeth.

Throughout we use the teeth, up and down teeth, apexes, bites, the
annular medial cycle $A(T)$, and the auxiliary plane graph $B(T)$
of~\cite{bauerfeld-medial-tire}.  Each tooth is a $3$-face of
$\mathsf{M}(T)$, and the inner faces of $B(T)$ (the root face and the
bite inner-gap faces) are the larger faces to be cut into teeth.

\begin{definition}[Walk-depth labelling and cut]
\label{def:walk-depth-cut}
Let $\mathsf{M}(T)$ be a full medial tire graph.  Assign walk depths and
cuts as follows.
\begin{enumerate}
  \item Pick an arbitrary up tooth, the \emph{entry tooth}.  It has walk
  depth $d$.
  \item Traverse all the teeth that bound the inner face incident to the
  entry tooth clockwise until we reach the entry tooth, incrementing the
  walk depth by $1$ for each tooth traversed.  (The \emph{inner face
  incident to the entry tooth} is the inner face of $B(T)$ whose boundary
  contains the annular edge of $A(T)$ carrying the entry tooth.)
  \item When you reach the last tooth in the face, perform a \emph{cut}
  by duplicating the annular vertex at which the traversal closes---the
  annular vertex of $A(T)$ shared by the last tooth and the entry tooth.
  \item Find the tooth $t$ with the highest walk depth which is a member
  of a bite.
  \item If $t$ is incident to a face $F$ with unlabelled teeth, traverse
  the teeth in $F$ starting from $t$ in the direction of the tooth
  incident to $t$ which is unlabelled, and increment the walk depth by
  $1$ as you travel.  (Here a tooth is \emph{incident to $t$} when it
  shares an annular vertex of $A(T)$ with $t$.)
  \item Repeat steps (3)--(5) until all teeth have been labelled.
\end{enumerate}
\end{definition}

\begin{remark}[Closing tooth of a descended face]
\label{rem:closing-tooth}
For the entry face the traversal of step (2) closes by returning to the
entry tooth, so the cut of step (3) duplicates the annular vertex shared
by the last tooth and the entry tooth.  For a face $F$ entered in step
(5), the traversal instead closes upon reaching an already-labelled
tooth: the other tooth of the bite through which $F$ was entered.  In
both cases the cut of step (3) duplicates the annular vertex shared by
the last newly labelled tooth and this \emph{closing tooth}.  Since both
teeth of a bite are labelled while traversing its parent face, every
descended face closes on such a tooth.
\end{remark}

\begin{example}[A worked walk-depth labelling and cut]
\label{ex:worked-cut}
Figure~\ref{fig:worked-cut} shows a full medial tire graph with annular
cycle of length $8$, generated by the full medial tire generator
of~\cite{bauerfeld-medial-tire}.  Its eight teeth are: three up teeth on
the annular edges $5,6,7$ in the root face; one bite pairing the annular
edges $0$ and $4$; and three singleton down teeth on the annular edges
$1,2,3$ lying in that bite's inner-gap face.

Take the up tooth on edge $5$ as the entry tooth, with walk depth $0$.
Its inner face is the root face, bounded by the teeth on edges
$5,6,7,0,4$ in clockwise order.  Step (2) labels them
\[
  5\mapsto 0,\quad 6\mapsto 1,\quad 7\mapsto 2,\quad
  0\mapsto 3,\quad 4\mapsto 4,
\]
and step (3) cuts by duplicating the annular vertex $a_5$ shared by the
last tooth (edge $4$) and the entry tooth (edge $5$).  The highest-depth
bite tooth is now the one on edge $4$ (walk depth $4$); it is incident to
the still-unlabelled inner-gap face of the bite $(0,4)$.  Entering that
face from edge $4$ toward its unlabelled neighbour, step (5) labels the
three down teeth
\[
  3\mapsto 5,\quad 2\mapsto 6,\quad 1\mapsto 7,
\]
and closes on the already-labelled bite tooth of edge $0$, so step (3)
cuts by duplicating the annular vertex $a_1$
(Remark~\ref{rem:closing-tooth}).  All eight teeth are now labelled, and
the two cuts leave only the outer face and the eight teeth as
$3$-faces.  The labelling and cuts are produced by the script
\texttt{experiments/medial\_tire\_cut\_labelling.py}.
\end{example}

\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.6,
    ann/.style={circle, fill=black, inner sep=1.0pt},
    upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},
    downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},
    bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},
    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},
    cut/.style={red!80!black, line width=1.3pt},
    cutlbl/.style={font=\tiny, text=red!75!black}]
\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;
\draw[tth] (-1.349,-0.559)--(-0.707,-0.707) (-1.349,-0.559)--(-1.000,-0.000);
\draw[tth] (-1.349,0.559)--(-1.000,-0.000) (-1.349,0.559)--(-0.707,0.707);
\draw[tth] (-0.559,1.349)--(-0.707,0.707) (-0.559,1.349)--(0.000,1.000);
\draw[tth] (0.554,0.230)--(0.707,0.707) (0.554,0.230)--(1.000,0.000);
\draw[tth] (0.554,-0.230)--(1.000,0.000) (0.554,-0.230)--(0.707,-0.707);
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\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
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\node[downv] at (0.230,-0.554) {};
\node[bitev] at (0.000,-0.000) {};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=1.6,
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    lbl/.style={font=\scriptsize},
    dlbl/.style={font=\scriptsize\bfseries, text=black},
    cut/.style={red!80!black, line width=1.3pt},
    cutlbl/.style={font=\tiny, text=red!75!black}]
\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;
\draw[tth] (-1.349,-0.559)--(-0.707,-0.707) (-1.349,-0.559)--(-1.000,-0.000);
\draw[tth] (-1.349,0.559)--(-1.000,-0.000) (-1.349,0.559)--(-0.707,0.707);
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\node[downv] at (0.554,-0.230) {};
\node[downv] at (0.230,-0.554) {};
\node[bitev] at (0.000,-0.000) {};
\node[dlbl] at (0.177,0.427) {3};
\node[dlbl] at (0.704,0.292) {7};
\node[dlbl] at (0.704,-0.292) {6};
\node[dlbl] at (0.292,-0.704) {5};
\node[dlbl] at (-0.177,-0.427) {4};
\node[dlbl] at (-1.101,-0.456) {0};
\node[dlbl] at (-1.101,0.456) {1};
\node[dlbl] at (-0.456,1.101) {2};
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\node[cutlbl] at (0.919,0.919) {cut 2};
\node[lbl, text=blue!60!black] at (-1.663,-0.689) {entry};
\end{tikzpicture}
\caption{A full medial tire graph (left) and its walk-depth labelling and
cut (right), from Example~\ref{ex:worked-cut}.  Black vertices are the
annular medial vertices of the cycle $A(T)$; blue vertices are up-tooth
apexes, red vertices are down-tooth apexes, and the larger red vertex is
the shared apex of the bite on annular edges $0$ and $4$.  On the right,
each tooth carries its walk depth, and the two red slits mark the cuts:
\emph{cut~1} duplicates $a_5$ as the root-face traversal closes, and
\emph{cut~2} duplicates $a_1$ as the bite's inner-gap face closes.  After
the cuts the only bounded faces are the eight teeth.}
\label{fig:worked-cut}
\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}