Add source cap cut to medial tire figures

papers/medial_tire_cuts/paper.tex

@@ -289,6 +289,13 @@ We label tread by tread, outward from the root:
   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
@@ -387,28 +394,35 @@ tooth carries its walk depth; the red slits are the two cuts.}
 \label{fig:real-cut}
 \end{figure}
 
-The same data sit inside the whole medial graph $M(G)$.
-Figure~\ref{fig:whole-medial} draws all of $M(G)$ for the graph of
-Example~\ref{ex:real-cut}, with the tread $T_2$ of
-Figure~\ref{fig:real-cut} highlighted in place: its annular medial cycle
-in black, its up and down teeth in blue and red carrying their walk
-depths, and the remaining medial vertices---those outside any recognised
-tire---in grey.  This is the assembled cut graph emitted by the
-experiment: every recognised tread contributes its cuts, and the tire
-pieces are glued to the rest of $M(G)$ along their boundary medial
-vertices.
+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_seed72.tikz}
-\caption{The whole medial graph $M(G)$ of the random maximal planar graph
-on $20$ vertices from Example~\ref{ex:real-cut}, with all tire cuts
-applied.  Grey vertices are medial vertices outside any recognised tire;
-the highlighted tread $T_2$ (cf.\ Figure~\ref{fig:real-cut}) has black
-annular medial vertices, blue up-tooth and red down-tooth apexes carrying
-their walk depths, and the larger red vertex is the bite apex.  Drawn by
-\texttt{experiments/draw\_medial\_tire\_cut.py} with the \texttt{--whole}
-option.}
+\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}