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Add walk-depth labelling/cut script and worked example

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New experiments/medial_tire_cut_labelling.py: takes a full medial tire
graph and an entry up tooth and runs the walk-depth labelling-and-cut
procedure, reusing the full medial tire generator's model and emitting
TikZ. Add a generator-produced 8-tooth example to the paper (Figure 1,
Example 2.3) showing the labelling and the two cuts, plus a remark
fixing the cut's closing tooth for descended faces.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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didericis
2026-06-14 22:00:52 -04:00
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@@ -115,6 +115,152 @@ cuts as follows.
\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);
\draw[tth] (0.230,-0.554)--(0.707,-0.707) (0.230,-0.554)--(0.000,-1.000);
\draw[tth] (0.000,-0.000)--(0.000,1.000) (0.000,-0.000)--(0.707,0.707);
\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
\node[ann] at (0.000,1.000) {};
\node[ann] at (0.707,0.707) {};
\node[ann] at (1.000,0.000) {};
\node[ann] at (0.707,-0.707) {};
\node[ann] at (0.000,-1.000) {};
\node[ann] at (-0.707,-0.707) {};
\node[ann] at (-1.000,-0.000) {};
\node[ann] at (-0.707,0.707) {};
\node[upv] at (-1.349,-0.559) {};
\node[upv] at (-1.349,0.559) {};
\node[upv] at (-0.559,1.349) {};
\node[downv] at (0.554,0.230) {};
\node[downv] at (0.554,-0.230) {};
\node[downv] at (0.230,-0.554) {};
\node[bitev] at (0.000,-0.000) {};
\end{tikzpicture}
\qquad
\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);
\draw[tth] (0.230,-0.554)--(0.707,-0.707) (0.230,-0.554)--(0.000,-1.000);
\draw[tth] (0.000,-0.000)--(0.000,1.000) (0.000,-0.000)--(0.707,0.707);
\draw[tth] (0.000,-0.000)--(0.000,-1.000) (0.000,-0.000)--(-0.707,-0.707);
\node[ann] at (0.000,1.000) {};
\node[ann] at (0.707,0.707) {};
\node[ann] at (1.000,0.000) {};
\node[ann] at (0.707,-0.707) {};
\node[ann] at (0.000,-1.000) {};
\node[ann] at (-0.707,-0.707) {};
\node[ann] at (-1.000,-0.000) {};
\node[ann] at (-0.707,0.707) {};
\node[upv] at (-1.349,-0.559) {};
\node[upv] at (-1.349,0.559) {};
\node[upv] at (-0.559,1.349) {};
\node[downv] at (0.554,0.230) {};
\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};
\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}
\begin{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}