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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.
  \item Finally, perform an apex cut at every up tooth except an entry
  tooth.  If the same medial vertex is the apex of two up teeth, do not
  cut that shared apex vertex.
\end{enumerate}
\end{definition}

\begin{remark}[Entry and shared up-apex exceptions]
\label{rem:up-apex-cut-exceptions}
For a single full medial tire graph there is one entry tooth.  In a
chained tire decomposition each tread has its own entry tooth, inherited
from the parent side or chosen at the root.  These entry triangles are
left uncut.  Shared up-apex vertices are also left uncut: the intended
cut set contains the apexes of singleton up teeth only.
\end{remark}

\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 closing cuts are followed by apex cuts at the non-entry up teeth on
edges $6$ and $7$.  The labelling and cuts are produced by the script
\texttt{lib/medial\_tire\_cut\_labelling.py}.
\end{example}

\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.6,
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    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}]
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\qquad
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\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};
\draw[cut] (-0.594,-0.594)--(-0.820,-0.820);
\node[cutlbl] at (-0.919,-0.919) {cut 1};
\draw[cut] (0.594,0.594)--(0.820,0.820);
\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}

\section{Chaining across the tire tree}

Definition~\ref{def:walk-depth-cut} labels and cuts a single full medial
tire graph.  We extend it to the whole medial graph $M(G)$ through the
medial tire decomposition of~\cite{bauerfeld-medial-tire}: the tire tree
decomposes $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one per
tread $T$, glued along their boundary medial vertices.  A parent tread's
inner level cycle is a child tread's outer level cycle, and the boundary
medial vertices on that shared cycle belong to both treads.

The key incidence is this.  A \emph{boundary} (singleton) down tooth of a
parent tread and the up tooth of the child tread glued to it across the
shared level cycle have the \emph{same apex}: both apexes are the same
medial vertex of $M(G)$, namely the medial vertex of an edge with both
endpoints on the shared level cycle.  We use this to carry the walk depth
from a parent into its children.

We label tread by tread, outward from the root:
\begin{itemize}
  \item a tread with no parent in the decomposition---in particular the
  innermost recognised tread---is treated as a \emph{root} and entered at
  an arbitrary up tooth with walk depth $0$;
  \item a child tread is entered at the up tooth whose apex is the parent's
  boundary down tooth of lowest walk depth; that entry up tooth's walk depth
  is one more than that down tooth's, and the walk then increments locally
  within the child as in Definition~\ref{def:walk-depth-cut}.
\end{itemize}

The source cap contributes one additional cut before the recognised
treads are assembled.  If the root tread enters at an up tooth whose apex
is the cap down tooth $xy$, we cut the cap annular vertex corresponding
to the counter-clockwise source edge incident to $xy$.  In the example of
Figure~\ref{fig:whole-medial}, the root entry apex is the cap down tooth
$14\!-\!4$, so the cap cut is placed at the medial vertex $14\!-\!5$.

\begin{remark}[Candidate down teeth for chaining]
\label{rem:chaining-candidates}
The down teeth eligible to fix a child's entry are exactly the
\emph{boundary} (singleton) down teeth of the parent: those lying in a
single tread face, whose apex is the shared boundary medial vertex glued to
a child up tooth.  A bite's two down teeth are \emph{not} eligible.  By the
definition of a bite in~\cite{bauerfeld-medial-tire} its annular edge borders
two tread faces, so a bite tooth is interior to the parent tread and its
apex is a boundary medial vertex of no child.  Hence ``the down tooth of
lowest walk depth'' is read among the boundary down teeth only; a bite of
even lower walk depth is skipped.
\end{remark}

Applying every tread's cuts to $M(G)$ assembles the per-tread labellings
and cuts into a single cut graph of $M(G)$ together with a global
walk-depth label map.  This pipeline---random maximal planar graph, medial
graph, tire decomposition at a vertex level source, and chained walk-depth
labelling and cut---is carried out by the experiment script
\texttt{experiments/run\_medial\_tire\_cut\_experiment.py}.

\begin{example}[A medial tire cut from a random graph]
\label{ex:real-cut}
Run on a random maximal planar graph on $20$ vertices (seed $72$, level
source vertex $9$), the experiment yields a single recognised tread
$T_2$, drawn in Figure~\ref{fig:real-cut} with the walk-depth labelling
and cut emitted by the graphics companion
\texttt{experiments/draw\_medial\_tire\_cut.py}.  Its annular cycle has
length $8$, with up teeth on annular edges $0,3,4$, singleton down teeth
on $1,6,7$, and a bite on the non-incident annular edges $2$ and $5$ (the
central shared apex).  Entering at the up tooth on edge $0$ with walk
depth $0$, the root face is labelled in order ($0,1,2$ then $3,4,5$) and
\emph{cut~1} duplicates $a_0$ as it closes; the walk then descends through
the bite into its inner-gap face, labelling the two teeth there ($6,7$),
and \emph{cut~2} duplicates $a_3$ as that face closes.  The two cuts leave
only the outer face and the eight teeth as $3$-faces.
\end{example}

\begin{figure}[h]
\centering
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    cutlbl/.style={font=\tiny, text=red!75!black}]
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\node[dlbl] at (-0.456,-1.101) {6};
\node[dlbl] at (-0.427,-0.336) {3};
\node[dlbl] at (-0.704,0.292) {4};
\node[dlbl] at (-0.292,0.704) {5};
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\node[cutlbl] at (0.000,1.300) {cut 1};
\draw[cut] (0.594,-0.594)--(0.820,-0.820);
\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}
\end{figure}

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}