coloring_nested_tire_graphs: switch bridge example to barbell (non-pendant bridge)

The previous bridge example used a pendant edge as the bridge.  A
pendant edge IS technically a bridge (single-edge cut), but the
intended notion was a "proper" non-pendant bridge: an edge cut
connecting two non-trivial subgraphs.  Replaced with the smallest
example:

  - B_out = 4-cycle on {0, 1, 2, 3}.
  - O = barbell on {4..9}: two disjoint triangles {4, 5, 6} and
    {7, 8, 9} connected by the bridge edge 6-7.
  - Annular triangulation by hand (12 triangles, all listed in the
    generator script with planarity verified by Sage).

The barbell case is structurally cleaner: BOTH endpoints of the
bridge have degree >= 2 in O, and the interior dual subgraph has
the two bridge-incident annular faces (d_5, d_6) as its trivalent
theta-graph vertices (in the pendant case, the trivalent vertices
were NOT bridge-incident, which was confusing).

Updates fig_partial_tire_dual_bridge.png and the figure caption.

Paper stays at 9 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual_bridge.py

@@ -1,17 +1,18 @@
 """Draw a partial tire dual D(T) for a tire whose inner outerplanar
-graph O has a bridge.
+graph O has a (non-pendant) bridge: a single edge whose removal
+disconnects O into two non-trivial components.
 
 Tire construction:
-  - Outer cycle B_out: triangle on {0, 1, 2}.
+  - Outer cycle B_out: 4-cycle on {0, 1, 2, 3}.
-  - Inner outerplanar O on {3, 4, 5, 6}: triangle 3-4-5 plus pendant
+  - Inner outerplanar O on {4..9}: two disjoint triangles {4, 5, 6}
-    edge 5-6 (the bridge of O).
+    and {7, 8, 9}, connected by the bridge edge 6-7.
-  - Annular triangulation with 8 triangles (computed by hand below).
+  - Annular triangulation built by hand below.
 
-The bridge 5-6 has both incident faces in the annular region, so in
+The bridge 6-7 has both incident faces in the annular region, so in
-the partial tire dual it contributes an interior dual edge (not a
+the partial tire dual it contributes an interior dual edge (not
-leaf).  This makes the interior dual subgraph a theta graph rather
+leaves).  This makes the interior dual subgraph a theta graph: two
-than a single cycle: two trivalent vertices (the two annular faces
+trivalent vertices (the two annular faces incident to the bridge,
-incident to the bridge) connected by three paths.
+or those with all interior edges) connected by three paths.
 """
 import math
 import os
@@ -21,54 +22,72 @@ from collections import defaultdict
 import matplotlib.pyplot as plt
 import matplotlib.patches as patches
 
+from sage.all import Graph
+
 
 # Explicit vertex positions
 POS = {
-    0: (0.0, 1.0),
+    0: (0.0, 1.6),         # outer top
-    1: (-0.866, -0.5),
+    1: (-1.6, 0.0),        # outer left
-    2: (0.866, -0.5),
+    2: (0.0, -1.6),        # outer bottom
-    3: (0.0, 0.32),
+    3: (1.6, 0.0),         # outer right
-    4: (-0.27, -0.17),
+    4: (-0.65, 0.42),      # left triangle, top
-    5: (0.13, -0.17),
+    5: (-0.65, -0.42),     # left triangle, bottom
-    6: (0.52, -0.07),
+    6: (-0.20, 0.0),       # left triangle, right (= bridge endpoint 1)
+    7: (0.20, 0.0),        # right triangle, left (= bridge endpoint 2)
+    8: (0.65, 0.42),       # right triangle, top
+    9: (0.65, -0.42),      # right triangle, bottom
 }
 
-# Annular triangles (in lattice-path-like order, but here just listed)
-TRIANGLES = [
-    (0, 1, 4),     # T1 -- O-move
-    (0, 4, 3),     # T2 -- has edge 4-3 boundary (inner triangle)
-    (1, 2, 5),     # T3 -- O-move
-    (1, 5, 4),     # T4 -- inner-triangle-edge boundary
-    (2, 0, 6),     # T5 -- O-move
-    (0, 3, 6),     # T6 -- ALL INTERIOR (bridge neighbour)
-    (3, 5, 6),     # T7 -- inner-triangle-edge boundary, contains bridge
-    (5, 2, 6),     # T8 -- ALL INTERIOR (bridge neighbour)
-]
-
-# Tire edges (all)
 T_EDGES = [
-    # outer cycle (B_out)
+    # outer cycle B_out (length 4)
-    (0, 1), (1, 2), (2, 0),
+    (0, 1), (1, 2), (2, 3), (3, 0),
-    # O: triangle 3-4-5
+    # O: triangle 4-5-6
-    (3, 4), (4, 5), (3, 5),
+    (4, 5), (5, 6), (4, 6),
-    # O: pendant / bridge
+    # O: triangle 7-8-9
-    (5, 6),
+    (7, 8), (8, 9), (7, 9),
-    # spokes
+    # O: bridge
-    (0, 3), (1, 4), (2, 5),
+    (6, 7),
-    # diagonals from annular triangulation
+    # annular edges (spokes + diagonals chosen to triangulate)
-    (0, 4), (1, 5), (0, 6), (2, 6), (3, 6),
+    (0, 4), (0, 6), (0, 8),
+    (1, 4), (1, 5),
+    (2, 5), (2, 6), (2, 7), (2, 9),
+    (3, 8), (3, 9),
+    (0, 7),       # extra diagonal to triangulate top
 ]
 
-# Boundary edges
+# Annular triangles -- listed explicitly so we know exactly which
-B_OUT_EDGES = [(0, 1), (1, 2), (2, 0)]
+# faces are which.  12 of them.
-# Edges of O on inner-triangle-face boundary (i.e., NOT the bridge)
+TRIANGLES = [
-B_IN_EDGES = [(3, 4), (4, 5), (3, 5)]
+    (0, 1, 4),     # top-left top
-# Bridge edge (interior to annulus)
+    (1, 4, 5),     # left side, with triangle 1
-BRIDGE = (5, 6)
+    (1, 2, 5),     # bottom-left
+    (2, 5, 6),     # right of left-triangle, bottom
+    (0, 4, 6),     # right of left-triangle, top
+    (0, 6, 7),     # top-center over the bridge (bridge-incident)
+    (2, 6, 7),     # bottom-center over the bridge (bridge-incident)
+    (0, 7, 8),     # right of right-triangle, top
+    (2, 7, 9),     # right of right-triangle, bottom
+    (0, 8, 3),     # top-right
+    (3, 8, 9),     # right side, with triangle 2
+    (2, 9, 3),     # bottom-right
+]
+
+# Boundary edges:
+B_OUT_EDGES = [(0, 1), (1, 2), (2, 3), (3, 0)]
+# Edges of O on inner-triangle-face boundaries (= NOT the bridge):
+B_IN_EDGES = [(4, 5), (5, 6), (4, 6),
+              (7, 8), (8, 9), (7, 9)]
+BRIDGE = (6, 7)
 
 
 def main():
-    # Map: edge -> list of triangle indices containing it
+    # Sanity-check via sage that the graph is planar
+    G = Graph(T_EDGES)
+    assert G.is_planar(), "Constructed tire is not planar"
+    print(f"Tire graph: n={G.order()}, m={G.size()}, planar=True")
+
+    # Map: edge -> triangle indices containing it
     edge_to_tris = defaultdict(list)
     for i, t in enumerate(TRIANGLES):
         for e in (frozenset({t[0], t[1]}),
@@ -76,67 +95,84 @@ def main():
                   frozenset({t[0], t[2]})):
             edge_to_tris[e].append(i)
 
-    # Verify each B_out edge is in exactly 1 annular triangle
+    # Verify each B_out edge in exactly 1 annular triangle
     for u, v in B_OUT_EDGES:
         ts = edge_to_tris[frozenset({u, v})]
-        assert len(ts) == 1, f"B_out edge {u}-{v} in {ts}"
+        assert len(ts) == 1, f"B_out edge {u}-{v} in {ts} (expected 1)"
-
+    # Each B_in (non-bridge O edge) in exactly 1 annular triangle
-    # Each B_in (= non-bridge O edge) is in exactly 1 annular triangle
     for u, v in B_IN_EDGES:
         ts = edge_to_tris[frozenset({u, v})]
-        assert len(ts) == 1, f"B_in edge {u}-{v} in {ts}"
+        assert len(ts) == 1, f"B_in edge {u}-{v} in {ts} (expected 1)"
-
+    # Bridge in exactly 2 annular triangles
-    # Bridge is in exactly 2 annular triangles (interior to annulus)
     bts = edge_to_tris[frozenset(BRIDGE)]
     print(f"Bridge {BRIDGE} is in annular triangles {bts}")
     assert len(bts) == 2
 
-    # Interior dual subgraph: connect annular triangles sharing an edge
+    # Verify every edge of T is accounted for
+    seen_in_tris = set()
+    for t in TRIANGLES:
+        for e in (frozenset({t[0], t[1]}),
+                  frozenset({t[1], t[2]}),
+                  frozenset({t[0], t[2]})):
+            seen_in_tris.add(e)
+    all_T_edges = {frozenset(e) for e in T_EDGES}
+    # Each edge appears in at least one triangle EXCEPT it might be that
+    # boundary edges to non-annular faces are accounted for differently.
+    # Annular triangles use boundary edges (B_out and inner-triangle-of-O
+    # edges) plus interior annular edges.  Every edge of T should appear
+    # in at least one annular triangle EXCEPT the inner faces of O don't
+    # exist as separate annular triangles.  Wait, the inner triangle
+    # faces of O (= {4,5,6} and {7,8,9}) are themselves face boundaries
+    # of T but NOT in the annular region.  So edges of the inner
+    # triangles of O appear in exactly ONE annular triangle (across the
+    # boundary) + the inner triangle face on the other side.
+    # Check: every T edge should appear in at most 2 annular triangles
+    # (and at least 1 if it's incident to any annular face).
+    for e in all_T_edges:
+        n = len(edge_to_tris[e])
+        assert n in (1, 2), \
+            f"Edge {sorted(e)} appears in {n} annular triangles"
+
+    # Interior dual adjacencies
     n_tri = len(TRIANGLES)
     tri_adj = defaultdict(set)
     for e, ts in edge_to_tris.items():
         if len(ts) == 2:
             tri_adj[ts[0]].add(ts[1])
             tri_adj[ts[1]].add(ts[0])
-
     print("Interior dual subgraph degree sequence:")
     for i in range(n_tri):
-        print(f"  T{i+1} ({TRIANGLES[i]}): degree {len(tri_adj[i])}")
+        print(f"  T{i} {TRIANGLES[i]}: degree {len(tri_adj[i])}")
 
-    # Centroids of triangles for d_f positions
+    # Centroids of triangles (for d_f positions)
-    centroids = []
+    centroids = [((POS[t[0]][0] + POS[t[1]][0] + POS[t[2]][0]) / 3,
-    for t in TRIANGLES:
+                  (POS[t[0]][1] + POS[t[1]][1] + POS[t[2]][1]) / 3)
-        cx = (POS[t[0]][0] + POS[t[1]][0] + POS[t[2]][0]) / 3
+                 for t in TRIANGLES]
-        cy = (POS[t[0]][1] + POS[t[1]][1] + POS[t[2]][1]) / 3
-        centroids.append((cx, cy))
 
     # === Plot ===
     fig, ax = plt.subplots(figsize=(10, 9))
 
-    # Draw underlying tire edges faintly
+    outer_set = {0, 1, 2, 3}
-    outer_set = {0, 1, 2}
+    inner_set = {4, 5, 6, 7, 8, 9}
-    inner_set = {3, 4, 5, 6}
+    bridge_fs = frozenset(BRIDGE)
     for u, v in T_EDGES:
         x1, y1 = POS[u]; x2, y2 = POS[v]
-        if u in outer_set and v in outer_set:
+        if frozenset({u, v}) == bridge_fs:
-            color = '#a8c9e8'; lw = 2.0
+            color, lw = '#cc6677', 2.8     # bridge of O
+        elif u in outer_set and v in outer_set:
+            color, lw = '#a8c9e8', 2.0
         elif u in inner_set and v in inner_set:
-            # Distinguish bridge (pendant) from triangle edges
+            color, lw = '#e8a8a8', 2.0
-            if frozenset({u, v}) == frozenset(BRIDGE):
-                color = '#cc6677'; lw = 2.5  # darker red for bridge
-            else:
-                color = '#e8a8a8'; lw = 2.0
         else:
-            color = '#cccccc'; lw = 1.0
+            color, lw = '#cccccc', 1.0
         ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1)
 
-    # Tire vertices
+    for v in outer_set:
-    for v in [0, 1, 2]:
         x, y = POS[v]
-        ax.plot(x, y, 'o', color='#a8c9e8', markersize=15, zorder=2)
+        ax.plot(x, y, 'o', color='#a8c9e8', markersize=16, zorder=2)
         ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
                     fontsize=9, fontweight='bold', zorder=3)
-    for v in [3, 4, 5, 6]:
+    for v in inner_set:
         x, y = POS[v]
         ax.plot(x, y, 'o', color='#e8a8a8', markersize=13, zorder=2)
         ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
@@ -144,24 +180,22 @@ def main():
 
     DUAL_COLOR = '#7d3c98'
     LEAF_COLOR = '#e67e22'
-    BRIDGE_DUAL_COLOR = '#cc4444'   # color the bridge's dual edge differently
+    BRIDGE_DUAL_COLOR = '#cc4444'
 
     # Interior dual edges
     for i in range(n_tri):
         for j in tri_adj[i]:
-            if j <= i:
+            if j <= i: continue
-                continue
-            x1, y1 = centroids[i]; x2, y2 = centroids[j]
-            # Find shared edge to colour-code bridge edge
             ti_es = {frozenset({TRIANGLES[i][a], TRIANGLES[i][b]})
                      for a, b in [(0, 1), (1, 2), (0, 2)]}
             tj_es = {frozenset({TRIANGLES[j][a], TRIANGLES[j][b]})
                      for a, b in [(0, 1), (1, 2), (0, 2)]}
             shared = (ti_es & tj_es).pop()
-            if shared == frozenset(BRIDGE):
+            if shared == bridge_fs:
                 ec, lw = BRIDGE_DUAL_COLOR, 3.0
             else:
                 ec, lw = DUAL_COLOR, 2.0
+            x1, y1 = centroids[i]; x2, y2 = centroids[j]
             ax.plot([x1, x2], [y1, y2], color=ec, linewidth=lw, zorder=4)
 
     # Leaves for B_out edges
@@ -170,18 +204,15 @@ def main():
         midx = (POS[u][0] + POS[v][0]) / 2
         midy = (POS[u][1] + POS[v][1]) / 2
         d = math.sqrt(midx**2 + midy**2)
-        push = 1.20
+        push = 1.18
         lpos = (midx * push / d, midy * push / d)
         cx, cy = centroids[ti]
         ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR,
                 linewidth=1.5, linestyle='--', zorder=4)
-        ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11,
+        ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=10,
                 zorder=5)
-        ax.annotate(f"$\\ell^{{\\mathrm{{out}}}}_{{{u},{v}}}$", lpos,
-                    color='white', ha='center', va='center', fontsize=6,
-                    fontweight='bold', zorder=6)
 
-    # Leaves for non-bridge B_in edges (= the 3 inner triangle edges)
+    # Leaves for non-bridge edges of O (= inner triangle edges)
     for u, v in B_IN_EDGES:
         ti = edge_to_tris[frozenset({u, v})][0]
         midx = (POS[u][0] + POS[v][0]) / 2
@@ -190,30 +221,21 @@ def main():
         vx, vy = midx - cx, midy - cy
         norm = math.sqrt(vx*vx + vy*vy)
         offset = 0.10
-        if norm > 1e-9:
+        nx, ny = (vx / norm, vy / norm) if norm > 1e-9 else (0, -1)
-            nx, ny = vx / norm, vy / norm
-        else:
-            nx, ny = 0, -1
         lpos = (midx + nx * offset, midy + ny * offset)
         ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR,
                 linewidth=1.5, linestyle='--', zorder=4)
-        ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11,
+        ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=10,
                 zorder=5)
-        ax.annotate(f"$\\ell^{{\\mathrm{{in}}}}_{{{u},{v}}}$", lpos,
-                    color='white', ha='center', va='center', fontsize=6,
-                    fontweight='bold', zorder=6)
 
-    # Interior dual vertices d_f, highlighting the two degree-3
+    # Interior dual vertices, highlight degree-3 (trivalent)
-    # vertices in a darker color (these are the trivalent vertices
-    # of the theta graph that the interior dual forms).
     for ti, (cx, cy) in enumerate(centroids):
-        is_deg3 = (len(tri_adj[ti]) == 3)
+        deg = len(tri_adj[ti])
-        marker_color = '#5a2273' if is_deg3 else DUAL_COLOR
+        c = '#5a2273' if deg == 3 else DUAL_COLOR
-        ax.plot(cx, cy, 's', color=marker_color, markersize=15, zorder=5)
+        ax.plot(cx, cy, 's', color=c, markersize=14, zorder=5)
         ax.annotate(f"$d_{{{ti}}}$", (cx, cy), color='white', ha='center',
-                    va='center', fontsize=7, fontweight='bold', zorder=6)
+                    va='center', fontsize=6, fontweight='bold', zorder=6)
 
-    # Legend
     legend_items = [
         plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8',
                    markersize=12, label='$B_{\\mathrm{out}}$ vertex'),
@@ -223,13 +245,13 @@ def main():
                    label='$B_{\\mathrm{out}}$ edge'),
         plt.Line2D([], [], color='#e8a8a8', linewidth=2.0,
                    label='triangle edge of $O$'),
-        plt.Line2D([], [], color='#cc6677', linewidth=2.5,
+        plt.Line2D([], [], color='#cc6677', linewidth=2.8,
-                   label='bridge of $O$ (= pendant)'),
+                   label='bridge of $O$'),
         plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR,
-                   markersize=12, label='$d_f$ (degree 2 in interior dual)'),
+                   markersize=12, label='$d_f$ (deg 2 in interior dual)'),
         plt.Line2D([], [], marker='s', color='w', markerfacecolor='#5a2273',
                    markersize=12,
-                   label='$d_f$ (degree 3 -- theta-graph trivalent)'),
+                   label='$d_f$ (deg 3 -- trivalent theta-graph vertex)'),
         plt.Line2D([], [], marker='D', color='w', markerfacecolor=LEAF_COLOR,
                    markersize=10, label='leaf'),
         plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0,
@@ -242,16 +264,18 @@ def main():
     ax.legend(handles=legend_items, loc='upper left',
               bbox_to_anchor=(1.02, 1.0), fontsize=8, frameon=False)
 
-    ax.set_xlim(-1.30, 1.50)
+    ax.set_xlim(-2.10, 2.60)
-    ax.set_ylim(-1.20, 1.20)
+    ax.set_ylim(-1.90, 1.90)
     ax.set_aspect('equal')
     ax.axis('off')
     ax.set_title(
-        'Partial tire dual $D(T)$ when $O$ has a bridge\n'
+        '$D(T)$ when $O$ is a barbell (two triangles + one bridge edge)\n'
-        '$O$ = triangle $\\{3,4,5\\}$ + pendant edge $5$--$6$;\n'
+        '$O$ = triangle $\\{4,5,6\\}$ $\\cup$ bridge $6$--$7$ $\\cup$ '
-        'interior dual is a theta graph (2 deg-3 vertices + 3 paths), '
+        'triangle $\\{7,8,9\\}$;\n'
-        'not a cycle.',
+        'bridge is interior to the annulus, so it contributes an interior '
-        fontsize=11)
+        'dual edge (red).\n'
+        'Interior dual is a theta graph (2 trivalent $d_f$ + 3 paths).',
+        fontsize=10)
 
     HERE = os.path.dirname(os.path.abspath(__file__))
     out = os.path.join(HERE, 'partial_tire_dual_bridge.png')

papers/coloring_nested_tire_graphs/paper.aux

@@ -11,21 +11,21 @@
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm  {out}}$ or of $B_{\mathrm  {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.8\hbox {}.}}{4}{}\protected@file@percent }
 \newlabel{fig:partial-tire-dual-example}{{3}{4}}
 \newlabel{prop:partial-tire-dual-structure}{{1.8}{4}}
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge. Here $B_{\mathrm  {out}}$ is a triangle on $\{0,1,2\}$ and $O$ is a triangle $\{3,4,5\}$ with a pendant edge $5$--$6$ (the bridge of $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph  {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph (two trivalent vertices $d_5, d_7$ connected by three internally vertex-disjoint paths in $D(T)$). Leaves come only from $B_{\mathrm  {out}}$ ($n = 3$ leaves) and from the three non-bridge edges of $O$ (the three triangle edges of the inner triangle).}}{5}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm  {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph  {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm  {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent }
 \newlabel{fig:partial-tire-dual-bridge}{{4}{5}}
 \newlabel{prop:no-level-d-pinch}{{1.9}{5}}
 \citation{bauerfeld-pds}
 \newlabel{lem:tire-component}{{1.10}{6}}
 \citation{bauerfeld-pds}
-\newlabel{rem:tire-component-degenerate}{{1.11}{7}}
+\newlabel{rem:tire-component-degenerate}{{1.11}{8}}
 \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}}
 \newlabel{prop:edge-vertex-bijection}{{1.13}{8}}
-\newlabel{rem:edge-vertex-corollary}{{1.14}{8}}
 \bibcite{bauerfeld-pds}{1}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.7778pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
+\newlabel{rem:edge-vertex-corollary}{{1.14}{9}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
 \gdef \@abspage@last{9}

papers/coloring_nested_tire_graphs/paper.log

@@ -1,4 +1,4 @@
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papers/coloring_nested_tire_graphs/paper.tex

@@ -237,20 +237,23 @@ Proposition~\ref{prop:partial-tire-dual-structure}.}
 
 \begin{figure}[h]
 \centering
-\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual_bridge.png}
+\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png}
 \caption{Partial tire dual $D(T)$ when the inner outerplanar graph
-$O$ has a bridge.  Here $B_{\mathrm{out}}$ is a triangle on
+$O$ has a bridge --- here a non-trivial edge cut connecting two
-$\{0,1,2\}$ and $O$ is a triangle $\{3,4,5\}$ with a pendant edge
+disjoint triangles.  $B_{\mathrm{out}}$ is a $4$-cycle on
-$5$--$6$ (the bridge of $O$).  Because both faces incident to the
+$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together
+with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing
+the bridge disconnects $O$).  Because both faces incident to the
 bridge are annular triangles, the bridge contributes an
 \emph{interior dual edge} (highlighted in red) rather than two
 leaves; consequently the interior dual subgraph is no longer the
 single $(n+m)$-cycle of
 Proposition~\ref{prop:partial-tire-dual-structure}, but a theta
-graph (two trivalent vertices $d_5, d_7$ connected by three
+graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident
-internally vertex-disjoint paths in $D(T)$).  Leaves come only from
+annular faces) are joined by three internally vertex-disjoint paths
-$B_{\mathrm{out}}$ ($n = 3$ leaves) and from the three non-bridge
+in $D(T)$.  Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves)
-edges of $O$ (the three triangle edges of the inner triangle).}
+and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves,
+three for each triangle).}
 \label{fig:partial-tire-dual-bridge}
 \end{figure}