coloring_nested_tire_graphs: add bridge-example partial tire dual figure beneath Fig 3
Adds fig_partial_tire_dual_bridge.png beneath the existing partial-
tire-dual figure (Figure 3). The new figure shows a tire graph
whose inner outerplanar O has a bridge:
B_out = triangle on {0, 1, 2};
O = triangle {3, 4, 5} plus pendant edge 5-6 (the bridge);
annular triangulation with 8 triangles (constructed by hand).
Key contrast with the previous figure: because both faces incident
to the bridge are annular triangles, the bridge contributes an
INTERIOR DUAL EDGE rather than two leaves. Consequently the
interior dual subgraph is no longer a single (n+m)-cycle (as in
Prop 1.8 for spoke-only tires) but a theta graph: two trivalent
d_f vertices connected by three internally vertex-disjoint paths.
Leaves come only from B_out (3 of them) and the three non-bridge
triangle edges of O (the inner-triangle face boundary).
Adds experiments/draw_partial_tire_dual_bridge.py to generate the
figure.
Paper grows from 8 to 9 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -0,0 +1,264 @@
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"""Draw a partial tire dual D(T) for a tire whose inner outerplanar
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graph O has a bridge.
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Tire construction:
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- Outer cycle B_out: triangle on {0, 1, 2}.
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- Inner outerplanar O on {3, 4, 5, 6}: triangle 3-4-5 plus pendant
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edge 5-6 (the bridge of O).
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- Annular triangulation with 8 triangles (computed by hand below).
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The bridge 5-6 has both incident faces in the annular region, so in
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the partial tire dual it contributes an interior dual edge (not a
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leaf). This makes the interior dual subgraph a theta graph rather
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than a single cycle: two trivalent vertices (the two annular faces
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incident to the bridge) connected by three paths.
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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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from collections import defaultdict
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import matplotlib.pyplot as plt
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import matplotlib.patches as patches
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# Explicit vertex positions
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POS = {
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0: (0.0, 1.0),
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1: (-0.866, -0.5),
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2: (0.866, -0.5),
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3: (0.0, 0.32),
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4: (-0.27, -0.17),
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5: (0.13, -0.17),
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6: (0.52, -0.07),
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}
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# Annular triangles (in lattice-path-like order, but here just listed)
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TRIANGLES = [
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(0, 1, 4), # T1 -- O-move
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(0, 4, 3), # T2 -- has edge 4-3 boundary (inner triangle)
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(1, 2, 5), # T3 -- O-move
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(1, 5, 4), # T4 -- inner-triangle-edge boundary
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(2, 0, 6), # T5 -- O-move
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(0, 3, 6), # T6 -- ALL INTERIOR (bridge neighbour)
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(3, 5, 6), # T7 -- inner-triangle-edge boundary, contains bridge
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(5, 2, 6), # T8 -- ALL INTERIOR (bridge neighbour)
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]
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# Tire edges (all)
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T_EDGES = [
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# outer cycle (B_out)
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(0, 1), (1, 2), (2, 0),
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# O: triangle 3-4-5
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(3, 4), (4, 5), (3, 5),
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# O: pendant / bridge
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(5, 6),
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# spokes
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(0, 3), (1, 4), (2, 5),
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# diagonals from annular triangulation
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(0, 4), (1, 5), (0, 6), (2, 6), (3, 6),
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]
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# Boundary edges
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B_OUT_EDGES = [(0, 1), (1, 2), (2, 0)]
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# Edges of O on inner-triangle-face boundary (i.e., NOT the bridge)
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B_IN_EDGES = [(3, 4), (4, 5), (3, 5)]
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# Bridge edge (interior to annulus)
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BRIDGE = (5, 6)
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def main():
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# Map: edge -> list of triangle indices containing it
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edge_to_tris = defaultdict(list)
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for i, t in enumerate(TRIANGLES):
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for e in (frozenset({t[0], t[1]}),
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frozenset({t[1], t[2]}),
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frozenset({t[0], t[2]})):
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edge_to_tris[e].append(i)
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# Verify each B_out edge is in exactly 1 annular triangle
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for u, v in B_OUT_EDGES:
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ts = edge_to_tris[frozenset({u, v})]
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assert len(ts) == 1, f"B_out edge {u}-{v} in {ts}"
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# Each B_in (= non-bridge O edge) is in exactly 1 annular triangle
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for u, v in B_IN_EDGES:
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ts = edge_to_tris[frozenset({u, v})]
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assert len(ts) == 1, f"B_in edge {u}-{v} in {ts}"
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# Bridge is in exactly 2 annular triangles (interior to annulus)
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bts = edge_to_tris[frozenset(BRIDGE)]
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print(f"Bridge {BRIDGE} is in annular triangles {bts}")
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assert len(bts) == 2
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# Interior dual subgraph: connect annular triangles sharing an edge
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n_tri = len(TRIANGLES)
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tri_adj = defaultdict(set)
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for e, ts in edge_to_tris.items():
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if len(ts) == 2:
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tri_adj[ts[0]].add(ts[1])
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tri_adj[ts[1]].add(ts[0])
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print("Interior dual subgraph degree sequence:")
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for i in range(n_tri):
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print(f" T{i+1} ({TRIANGLES[i]}): degree {len(tri_adj[i])}")
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# Centroids of triangles for d_f positions
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centroids = []
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for t in TRIANGLES:
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cx = (POS[t[0]][0] + POS[t[1]][0] + POS[t[2]][0]) / 3
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cy = (POS[t[0]][1] + POS[t[1]][1] + POS[t[2]][1]) / 3
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centroids.append((cx, cy))
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# === Plot ===
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fig, ax = plt.subplots(figsize=(10, 9))
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# Draw underlying tire edges faintly
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outer_set = {0, 1, 2}
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inner_set = {3, 4, 5, 6}
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for u, v in T_EDGES:
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x1, y1 = POS[u]; x2, y2 = POS[v]
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if u in outer_set and v in outer_set:
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color = '#a8c9e8'; lw = 2.0
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elif u in inner_set and v in inner_set:
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# Distinguish bridge (pendant) from triangle edges
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if frozenset({u, v}) == frozenset(BRIDGE):
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color = '#cc6677'; lw = 2.5 # darker red for bridge
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else:
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color = '#e8a8a8'; lw = 2.0
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else:
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color = '#cccccc'; lw = 1.0
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ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1)
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# Tire vertices
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for v in [0, 1, 2]:
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x, y = POS[v]
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ax.plot(x, y, 'o', color='#a8c9e8', markersize=15, zorder=2)
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ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
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fontsize=9, fontweight='bold', zorder=3)
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for v in [3, 4, 5, 6]:
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x, y = POS[v]
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ax.plot(x, y, 'o', color='#e8a8a8', markersize=13, zorder=2)
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ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
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fontsize=8, fontweight='bold', zorder=3)
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DUAL_COLOR = '#7d3c98'
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LEAF_COLOR = '#e67e22'
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BRIDGE_DUAL_COLOR = '#cc4444' # color the bridge's dual edge differently
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# Interior dual edges
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for i in range(n_tri):
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for j in tri_adj[i]:
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if j <= i:
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continue
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x1, y1 = centroids[i]; x2, y2 = centroids[j]
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# Find shared edge to colour-code bridge edge
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ti_es = {frozenset({TRIANGLES[i][a], TRIANGLES[i][b]})
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for a, b in [(0, 1), (1, 2), (0, 2)]}
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tj_es = {frozenset({TRIANGLES[j][a], TRIANGLES[j][b]})
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for a, b in [(0, 1), (1, 2), (0, 2)]}
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shared = (ti_es & tj_es).pop()
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if shared == frozenset(BRIDGE):
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ec, lw = BRIDGE_DUAL_COLOR, 3.0
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else:
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ec, lw = DUAL_COLOR, 2.0
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ax.plot([x1, x2], [y1, y2], color=ec, linewidth=lw, zorder=4)
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# Leaves for B_out edges
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for u, v in B_OUT_EDGES:
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ti = edge_to_tris[frozenset({u, v})][0]
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midx = (POS[u][0] + POS[v][0]) / 2
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midy = (POS[u][1] + POS[v][1]) / 2
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d = math.sqrt(midx**2 + midy**2)
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push = 1.20
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lpos = (midx * push / d, midy * push / d)
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cx, cy = centroids[ti]
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ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR,
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linewidth=1.5, linestyle='--', zorder=4)
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ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11,
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zorder=5)
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ax.annotate(f"$\\ell^{{\\mathrm{{out}}}}_{{{u},{v}}}$", lpos,
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color='white', ha='center', va='center', fontsize=6,
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fontweight='bold', zorder=6)
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# Leaves for non-bridge B_in edges (= the 3 inner triangle edges)
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for u, v in B_IN_EDGES:
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ti = edge_to_tris[frozenset({u, v})][0]
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midx = (POS[u][0] + POS[v][0]) / 2
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midy = (POS[u][1] + POS[v][1]) / 2
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cx, cy = centroids[ti]
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vx, vy = midx - cx, midy - cy
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norm = math.sqrt(vx*vx + vy*vy)
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offset = 0.10
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if norm > 1e-9:
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nx, ny = vx / norm, vy / norm
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else:
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nx, ny = 0, -1
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lpos = (midx + nx * offset, midy + ny * offset)
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ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR,
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linewidth=1.5, linestyle='--', zorder=4)
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ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11,
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zorder=5)
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ax.annotate(f"$\\ell^{{\\mathrm{{in}}}}_{{{u},{v}}}$", lpos,
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color='white', ha='center', va='center', fontsize=6,
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fontweight='bold', zorder=6)
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# Interior dual vertices d_f, highlighting the two degree-3
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# vertices in a darker color (these are the trivalent vertices
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# of the theta graph that the interior dual forms).
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for ti, (cx, cy) in enumerate(centroids):
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is_deg3 = (len(tri_adj[ti]) == 3)
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marker_color = '#5a2273' if is_deg3 else DUAL_COLOR
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ax.plot(cx, cy, 's', color=marker_color, markersize=15, zorder=5)
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ax.annotate(f"$d_{{{ti}}}$", (cx, cy), color='white', ha='center',
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va='center', fontsize=7, fontweight='bold', zorder=6)
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# Legend
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legend_items = [
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plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8',
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markersize=12, label='$B_{\\mathrm{out}}$ vertex'),
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plt.Line2D([], [], marker='o', color='w', markerfacecolor='#e8a8a8',
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markersize=11, label='$V(O)$ vertex'),
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plt.Line2D([], [], color='#a8c9e8', linewidth=2.0,
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label='$B_{\\mathrm{out}}$ edge'),
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plt.Line2D([], [], color='#e8a8a8', linewidth=2.0,
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label='triangle edge of $O$'),
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plt.Line2D([], [], color='#cc6677', linewidth=2.5,
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label='bridge of $O$ (= pendant)'),
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plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR,
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markersize=12, label='$d_f$ (degree 2 in interior dual)'),
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plt.Line2D([], [], marker='s', color='w', markerfacecolor='#5a2273',
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markersize=12,
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label='$d_f$ (degree 3 -- theta-graph trivalent)'),
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plt.Line2D([], [], marker='D', color='w', markerfacecolor=LEAF_COLOR,
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markersize=10, label='leaf'),
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plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0,
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label='interior dual edge'),
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plt.Line2D([], [], color=BRIDGE_DUAL_COLOR, linewidth=3.0,
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label='dual of the bridge'),
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plt.Line2D([], [], color=LEAF_COLOR, linewidth=1.5, linestyle='--',
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label='leaf edge'),
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]
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ax.legend(handles=legend_items, loc='upper left',
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bbox_to_anchor=(1.02, 1.0), fontsize=8, frameon=False)
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ax.set_xlim(-1.30, 1.50)
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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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ax.set_title(
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'Partial tire dual $D(T)$ when $O$ has a bridge\n'
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'$O$ = triangle $\\{3,4,5\\}$ + pendant edge $5$--$6$;\n'
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'interior dual is a theta graph (2 deg-3 vertices + 3 paths), '
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'not a cycle.',
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fontsize=11)
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HERE = os.path.dirname(os.path.abspath(__file__))
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out = os.path.join(HERE, 'partial_tire_dual_bridge.png')
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plt.savefig(out, dpi=160, bbox_inches='tight')
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plt.close()
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print(f"wrote {out}")
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if __name__ == '__main__':
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main()
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@@ -11,19 +11,21 @@
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\@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 }
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\newlabel{fig:partial-tire-dual-example}{{3}{4}}
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\newlabel{prop:partial-tire-dual-structure}{{1.8}{4}}
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\@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 }
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\newlabel{fig:partial-tire-dual-bridge}{{4}{5}}
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\newlabel{prop:no-level-d-pinch}{{1.9}{5}}
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\newlabel{lem:tire-component}{{1.10}{5}}
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\citation{bauerfeld-pds}
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\newlabel{lem:tire-component}{{1.10}{6}}
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\citation{bauerfeld-pds}
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\newlabel{rem:tire-component-degenerate}{{1.11}{7}}
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\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
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\newlabel{prop:edge-vertex-bijection}{{1.13}{7}}
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\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}}
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\newlabel{prop:edge-vertex-bijection}{{1.13}{8}}
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\newlabel{rem:edge-vertex-corollary}{{1.14}{8}}
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\bibcite{bauerfeld-pds}{1}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{12.7778pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\newlabel{rem:edge-vertex-corollary}{{1.14}{8}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
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\gdef \@abspage@last{8}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
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\gdef \@abspage@last{9}
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@@ -1,4 +1,4 @@
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 19:52
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 20:43
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entering extended mode
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@@ -213,36 +213,51 @@ File: fig_partial_tire_dual.png Graphic file (type png)
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<use fig_partial_tire_dual.png>
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Package pdftex.def Info: fig_partial_tire_dual.png used on input line 225.
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[4 <./fig_partial_tire_dual.png>] [5] [6] [7] [8] (./paper.aux) )
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<fig_partial_tire_dual_bridge.png, id=31, 765.6103pt x 542.47668pt>
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File: fig_partial_tire_dual_bridge.png Graphic file (type png)
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<use fig_partial_tire_dual_bridge.png>
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Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 2
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LaTeX Warning: `h' float specifier changed to `ht'.
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[4 <./fig_partial_tire_dual.png>] [5 <./fig_partial_tire_dual_bridge.png>]
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[6]
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[7]
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[8] [9] (./paper.aux) )
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Output written on paper.pdf (8 pages, 622907 bytes).
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69i,8n,76p,907b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s
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69 compressed objects within 1 object stream
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123 PDF objects out of 1000 (max. 8388607)
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71 compressed objects within 1 object stream
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16 words of extra memory for PDF output out of 10000 (max. 10000000)
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21 words of extra memory for PDF output out of 10000 (max. 10000000)
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Binary file not shown.
@@ -235,6 +235,25 @@ Proposition~\ref{prop:partial-tire-dual-structure}.}
|
||||
\label{fig:partial-tire-dual-example}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.78\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
|
||||
$\{0,1,2\}$ 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~\ref{prop:partial-tire-dual-structure}, 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).}
|
||||
\label{fig:partial-tire-dual-bridge}
|
||||
\end{figure}
|
||||
|
||||
\begin{proposition}[Structure of $D(T)$ when the annular triangulation
|
||||
is spoke-only]
|
||||
\label{prop:partial-tire-dual-structure}
|
||||
|
||||
Reference in New Issue
Block a user