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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{thebibliography}{9}

\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.

\end{thebibliography}

\end{document}
