coloring_nested_tire_graphs: extend Def 1.5 with degenerate boundaries; add tire-component lemma + example figure
- Extends Definition 1.5 (tire graph) to allow either B_out or B_in to
be a single vertex (a "degenerate" boundary); the tire is then a
triangulated disk with that vertex as apex.
- Adds Remark 1.6 with vertex/edge/triangle counts for both the
non-degenerate (annulus) and degenerate (disk) cases.
- Adds Lemma 1.7 (tire-component lemma): for a maximal planar graph G
with level source S and depth d ≥ 0, every connected component C' of
the dual subgraph on dual vertices of depth d induces a subgraph C
on G that is a tire graph; its two boundary parts are the level-d
and level-(d+1) induced subgraphs (in either order), and its
triangular faces are exactly the faces of G in F_{C'}.
- Adds Remark 1.8 noting the d=0 degenerate-source case as an example.
- Adds a figure (fig_tire_example.png) illustrating the definition with
m=6 outer cycle, k=4 inner cycle, one chord in O, plus a legend
identifying B_out, B_in, O's chord, and E_ann.
- Adds experiments/tire_def_figure.py to regenerate the figure.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
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"""Generate the example figure for Definition 1.5 (tire graph) of the
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paper. Produces fig_tire_example.png at the paper's top level.
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Picks a specific small tire (m=6 outer cycle, k=4 inner cycle with one
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chord) so that all four named parts B_out, O, B_in, E_ann are visible
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and the annular triangles are individually legible.
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"""
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import math
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import os
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import sys
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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import matplotlib.pyplot as plt
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import matplotlib.patches as patches
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from tire_graph import random_tire, planar_positions
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def draw_tire_def(tire, filename):
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m, k = tire['m'], tire['k']
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outer, inner = tire['outer'], tire['inner']
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edges = tire['edges']
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R_out, R_in = 1.0, 0.45
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pos = planar_positions(tire, R_out=R_out, R_in=R_in)
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fig, ax = plt.subplots(figsize=(6.5, 6.5))
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# guide circles for reference
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for r in (R_out + 0.04, R_in - 0.04):
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ax.add_patch(patches.Circle((0, 0), r, fill=False,
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edgecolor='lightgray',
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linewidth=0.5, linestyle='--'))
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outer_set = set(outer)
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inner_set = set(inner)
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C = {
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'outer_cycle': '#1f77b4',
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'inner_cycle': '#d62728',
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'inner_chord': '#ff7f0e',
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'spoke': '#7f7f7f',
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}
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# classify and draw edges
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for (u, v) in edges:
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if u in outer_set and v in outer_set:
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color, lw, label = C['outer_cycle'], 2.6, 'B_out'
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elif u in inner_set and v in inner_set:
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ia, ib = u - m, v - m
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d = abs(ia - ib)
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d = min(d, k - d)
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if d == 1:
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color, lw, label = C['inner_cycle'], 2.6, 'B_in'
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else:
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color, lw, label = C['inner_chord'], 1.8, 'O chord'
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else:
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color, lw, label = C['spoke'], 1.1, 'E_ann'
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x1, y1 = pos[u]; x2, y2 = pos[v]
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ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1)
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# vertices
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for v in outer:
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x, y = pos[v]
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ax.plot(x, y, 'o', color=C['outer_cycle'], markersize=14, zorder=2)
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ax.annotate(str(v), (x, y), color='white', ha='center',
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va='center', fontsize=9, fontweight='bold', zorder=3)
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for v in inner:
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x, y = pos[v]
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ax.plot(x, y, 'o', color=C['inner_cycle'], markersize=13, zorder=2)
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ax.annotate(str(v), (x, y), color='white', ha='center',
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va='center', fontsize=8, fontweight='bold', zorder=3)
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# legend
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legend_items = [
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plt.Line2D([], [], color=C['outer_cycle'], linewidth=2.6,
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label=r'$B_{\mathrm{out}}$ (outer boundary, $m=6$)'),
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plt.Line2D([], [], color=C['inner_cycle'], linewidth=2.6,
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label=r'$B_{\mathrm{in}}$ (inner boundary, $k=4$)'),
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plt.Line2D([], [], color=C['inner_chord'], linewidth=1.8,
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label=r'chord of $O$'),
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plt.Line2D([], [], color=C['spoke'], linewidth=1.1,
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label=r'$E_{\mathrm{ann}}$ (annular edges)'),
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]
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ax.legend(handles=legend_items, loc='upper left',
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bbox_to_anchor=(1.0, 1.0), fontsize=10, frameon=False)
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ax.set_xlim(-1.20, 1.20)
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ax.set_ylim(-1.20, 1.20)
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ax.set_aspect('equal')
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ax.axis('off')
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plt.savefig(filename, dpi=160, bbox_inches='tight')
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plt.close()
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def main():
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# m=6 outer, k=4 inner, 1 chord — try a few seeds to pick a clean one
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paper_dir = os.path.abspath(os.path.join(HERE, '..'))
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candidates = [(6, 4, 1, s) for s in (3, 5, 8, 13, 21)]
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# Use seed=3 as the chosen example (good lattice-path balance)
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for (m, k, nc, seed) in candidates[:1]:
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tire = random_tire(m, k, n_chords=nc, seed=seed)
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fn = os.path.join(paper_dir, 'fig_tire_example.png')
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draw_tire_def(tire, fn)
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print(f" wrote {fn}")
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print(f" m={m}, k={k}, chords={tire['inner_chords']}, "
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f"path={tire['lattice_path']}")
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if __name__ == '__main__':
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main()
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@@ -1,12 +1,17 @@
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\relax
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\newlabel{def:dual-depth}{{1.4}{1}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
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\newlabel{fig:dual-depth}{{1}{2}}
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\newlabel{def:tire-graph}{{1.5}{2}}
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\newlabel{rem:tire-counts}{{1.6}{2}}
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\newlabel{tocindent-1}{0pt}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
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\newlabel{fig:dual-depth}{{1}{2}}
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\newlabel{def:tire-graph}{{1.5}{2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent }
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\newlabel{fig:tire-example}{{2}{3}}
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\newlabel{lem:tire-component}{{1.7}{3}}
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\newlabel{rem:tire-component-degenerate}{{1.8}{3}}
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@@ -116,39 +116,94 @@ vertex.}
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\begin{definition}[Tire graph]
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\label{def:tire-graph}
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Let $C_{\mathrm{out}}$ be a simple cycle of length $m \geq 3$, and let
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$O$ be an outerplanar graph whose outer-face boundary $C_{\mathrm{in}}$
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is a simple cycle of length $k \geq 3$, with $V(C_{\mathrm{out}}) \cap
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V(O) = \emptyset$. A \emph{tire graph} on $(C_{\mathrm{out}}, O)$ is a
|
||||
plane graph $T$ with
|
||||
A \emph{tire graph} consists of a plane graph $T$ together with two
|
||||
\emph{boundary parts} $B_{\mathrm{out}}, B_{\mathrm{in}} \subseteq T$
|
||||
and an \emph{inner outerplanar graph} $O \subseteq T$, where each of
|
||||
$B_{\mathrm{out}}$ and the outer-face boundary $B_{\mathrm{in}}$ of $O$
|
||||
is either
|
||||
\begin{itemize}
|
||||
\item a simple cycle of length $\geq 3$, or
|
||||
\item a single vertex (a \emph{degenerate} boundary),
|
||||
\end{itemize}
|
||||
with at most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ degenerate, and
|
||||
$V(B_{\mathrm{out}}) \cap V(O) = \emptyset$. The vertex and edge sets
|
||||
of $T$ are
|
||||
\[
|
||||
V(T) = V(C_{\mathrm{out}}) \cup V(O),
|
||||
V(T) = V(B_{\mathrm{out}}) \cup V(O),
|
||||
\qquad
|
||||
E(T) = E(C_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
|
||||
E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
|
||||
\]
|
||||
where $E_{\mathrm{ann}}$ is a set of edges --- the \emph{annular edges}
|
||||
--- such that, in the plane embedding of $T$, the closed annulus with
|
||||
outer boundary $C_{\mathrm{out}}$ and inner boundary $C_{\mathrm{in}}$
|
||||
is partitioned into triangular faces. Equivalently, the bounded faces
|
||||
of $T$ that are not faces of $O$ are all triangles, and together they
|
||||
tile the annular region between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$.
|
||||
where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
|
||||
property that, in the plane embedding of $T$, the closed planar region
|
||||
$R$ bounded externally by $B_{\mathrm{out}}$ and internally by
|
||||
$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose
|
||||
union is $R$. The region $R$ is a closed annulus when both
|
||||
$B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ are cycles, and a closed disk
|
||||
when exactly one of them is a single vertex.
|
||||
|
||||
We call $C_{\mathrm{out}}$ the \emph{outer cycle}, $O$ the \emph{inner
|
||||
outerplanar graph}, and $C_{\mathrm{in}}$ the \emph{inner cycle} of
|
||||
$T$. When $O = C_{\mathrm{in}}$ (the inner outerplanar graph has no
|
||||
chords), $T$ is a tire graph \emph{with empty inner}; in general $O$
|
||||
contributes only chords inside the disk bounded by $C_{\mathrm{in}}$
|
||||
and does not interact with $E_{\mathrm{ann}}$.
|
||||
We call $B_{\mathrm{out}}$ the \emph{outer boundary}, $O$ the
|
||||
\emph{inner outerplanar graph}, and $B_{\mathrm{in}}$ the \emph{inner
|
||||
boundary} of $T$. A tire graph in which $B_{\mathrm{out}}$
|
||||
(respectively $B_{\mathrm{in}}$) is a single vertex is said to have a
|
||||
\emph{degenerate outer (respectively inner) boundary}; in either case
|
||||
$T$ is a triangulated closed disk with that vertex as apex.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.78\textwidth]{fig_tire_example.png}
|
||||
\caption{A tire graph with non-degenerate boundaries: outer boundary
|
||||
$B_{\mathrm{out}}$ a $6$-cycle on vertices $0,\dots,5$ (blue), inner
|
||||
boundary $B_{\mathrm{in}}$ a $4$-cycle on vertices $6,\dots,9$ (red),
|
||||
inner outerplanar graph $O = B_{\mathrm{in}} \cup \{7\text{--}9\}$
|
||||
(with one chord, orange), and $E_{\mathrm{ann}}$ (grey) tiling the
|
||||
annulus between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ by ten
|
||||
triangular faces.}
|
||||
\label{fig:tire-example}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
A tire graph on $(C_{\mathrm{out}}, O)$ has $|V(C_{\mathrm{out}})| +
|
||||
|V(O)| = m + k$ vertices, exactly $m + k$ annular triangles
|
||||
in the annulus between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$ (by
|
||||
Euler's formula on the annulus), and exactly $m + k$ annular edges
|
||||
in $E_{\mathrm{ann}}$, of which the $m + k$ triangles share their
|
||||
three edges with the boundaries $E(C_{\mathrm{out}}) \cup
|
||||
E(C_{\mathrm{in}})$ and with each other.
|
||||
\label{rem:tire-counts}
|
||||
Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By
|
||||
Euler's formula on the annular (resp.\ disk) region $R$, the tire graph
|
||||
has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
|
||||
annular edges when neither boundary is degenerate; when exactly one
|
||||
boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
|
||||
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
|
||||
\end{remark}
|
||||
|
||||
\begin{lemma}[Tire-component lemma]
|
||||
\label{lem:tire-component}
|
||||
Let $G$ be a maximal planar graph with fixed embedding $\Pi_G$ and let
|
||||
$S \subseteq V(G)$ be a level source. For $d \geq 0$, let
|
||||
\[
|
||||
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
|
||||
\]
|
||||
be the inner-dual subgraph on dual vertices of dual depth $d$, and let
|
||||
$C'$ be a connected component of $G'_d$. Write
|
||||
$F_{C'} := \{f : d_f \in V(C')\}$,
|
||||
$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and
|
||||
$C := G[V_{C'}]$ with the embedding inherited from $\Pi_G$.
|
||||
|
||||
Then $C$, together with its inherited embedding, is a tire graph in the
|
||||
sense of Definition~\ref{def:tire-graph}: the two boundary parts
|
||||
$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ of $C$ are the level-$d$
|
||||
subgraph $G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph
|
||||
$G[V_{C'} \cap L_{d+1}]$, in either order, and the triangular faces of
|
||||
$C$ inside its closed boundary region are exactly the faces of $G$ in
|
||||
$F_{C'}$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:tire-component-degenerate}
|
||||
Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
|
||||
degenerate. For instance, at $d = 0$ with single-vertex source
|
||||
$S = \{v_0\}$ the unique component of $G'_0$ has
|
||||
$V_{C'} \cap L_0 = \{v_0\}$ --- the degenerate boundary --- and
|
||||
$V_{C'} \cap L_1$ a cycle (the link of $v_0$ in $G$). Which of the two
|
||||
parts is $B_{\mathrm{out}}$ and which is $B_{\mathrm{in}}$ depends on
|
||||
the orientation of the inherited embedding (equivalently, on which side
|
||||
of $C$ contains the rest of $\Pi_G$).
|
||||
\end{remark}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user