Add medial tire cut experiment and chaining section

New experiments/run_medial_tire_cut_experiment.py: generates a random
maximal planar graph (stacked seed + random diagonal flips), builds the
medial graph, takes the tire decomposition at a random vertex level
source, walk-depth labels and cuts each full medial tire graph chained
down the tire tree, and assembles one final cut graph of M(G) with a
global label map (data only; graphics go in a separate script).

Fix label_and_cut: the root face is None, which collided with the
next(..., None) sentinel, leaving teeth unlabelled when the entry up
tooth lay inside a bite gap; use a distinct sentinel so the ascent to
the root face runs.

Add a "Chaining across the tire tree" section to the paper, clarifying
that the candidate parent down teeth are the boundary (singleton) down
teeth only -- bite teeth are interior to the parent and shared with no
child, so a lower-walk bite is skipped.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_cuts/paper.tex

@@ -261,6 +261,54 @@ 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}
+
+\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{thebibliography}{9}
 
 \bibitem{bauerfeld-medial-tire}