dual_decomposition: reduced-dual definition, verification, and step figures
Add Definition 2.1 (reduced dual) and a remark on cubicity/planarity, plus an experiment verifying it on the icosahedron/dodecahedron and four figures, one per construction step. reduced_dual.py builds G' = dodecahedron (dual of the icosahedron), applies the construction, and confirms the result is a cubic, planar, simple graph whose dual is a simple triangulation. Finding: the construction is an n -> n-2 reduction (12 -> 10 here), not n-1, since the single apex v_n collapses one more vertex than a standard pentagon re-triangulation; the result also re-introduces degree-3 and degree-4 vertices (degree seq [7,5,5,5,5,5,5,4,4,3]). draw_reduced_dual_steps.py renders fig_reduced_dual_step1..4.png, embedded as a 2x2 grid after the definition. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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"""Draw the four steps of the reduced-dual construction (Definition 2.1).
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Uses the dodecahedron G' = dual of the icosahedron, with F_v the inner pentagon,
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as built in reduced_dual.py. Produces fig_reduced_dual_step{1..4}.png.
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"""
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import os
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import math
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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from matplotlib.lines import Line2D
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from reduced_dual import build_dual, apply_reduction
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OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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GRAY = '#9ca3af'
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DARK = '#374151'
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GHOST = '#fca5a5'
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DEG2 = '#f59e0b'
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APEX = '#16a34a'
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CHORD = '#2563eb'
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FACE = '#fef9c3'
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def draw_edges(ax, G, pos, nodes=None, **kw):
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for u, v in G.edges():
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if nodes is not None and (u not in nodes or v not in nodes):
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continue
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(x0, y0), (x1, y1) = pos[u], pos[v]
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ax.plot([x0, x1], [y0, y1], **kw)
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def draw_nodes(ax, pos, nodes, **kw):
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xs = [pos[v][0] for v in nodes]
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ys = [pos[v][1] for v in nodes]
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ax.scatter(xs, ys, **kw)
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def face_F_polygon(pos):
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"""The new central face F: decagon alternating b_i, c_i clockwise."""
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order = []
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for i in range(5):
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order += [('b', i), ('c', i)]
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return [pos[v] for v in order]
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def base_canvas(title):
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fig, ax = plt.subplots(figsize=(8.5, 8.5))
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ax.set_aspect('equal')
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ax.axis('off')
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ax.set_title(title, fontsize=12)
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return fig, ax
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def main():
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Gp, pos, Fv = build_dual()
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res = apply_reduction(Gp, pos, Fv, i=0)
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Ghat, npos, A = res['Ghat'], res['pos'], res['A']
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v_n, apex_nbrs, chord = res['v_n'], res['apex_nbrs'], res['chord']
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survivors = [v for v in Gp if v not in Fv] # b, c, d families
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surv_set = set(survivors)
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deg2 = list(A) # the five b_i
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# surviving edges (both endpoints survive) vs deleted edges (touch an a_i)
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surv_edges = [(u, v) for u, v in Gp.edges()
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if u in surv_set and v in surv_set]
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del_edges = [(u, v) for u, v in Gp.edges()
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if u not in surv_set or v not in surv_set]
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def draw_surviving(ax):
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ax.add_patch(Polygon(face_F_polygon(pos), closed=True,
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facecolor=FACE, edgecolor='none', zorder=0))
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for u, v in surv_edges:
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(x0, y0), (x1, y1) = pos[u], pos[v]
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ax.plot([x0, x1], [y0, y1], color=GRAY, lw=1.6, zorder=1)
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others = [v for v in survivors if v not in deg2]
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draw_nodes(ax, pos, others, s=120, color=DARK, zorder=3)
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def draw_ghosts(ax):
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for u, v in del_edges:
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(x0, y0), (x1, y1) = pos[u], pos[v]
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ax.plot([x0, x1], [y0, y1], color=GHOST, lw=1.2, ls='--', zorder=1)
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draw_nodes(ax, pos, Fv, s=120, color='white', edgecolors=GHOST,
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linewidths=1.5, zorder=2)
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for v in Fv:
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ax.plot(*pos[v], marker='x', color=GHOST, ms=8, zorder=3)
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# ----- Step 1: delete F_v's boundary; five degree-2 vertices on face F -----
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fig, ax = base_canvas(
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"Step 1: delete the five dual vertices on $\\partial F_v$.\n"
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"Their outer neighbours drop to degree 2 (orange) and lie on a new "
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"face $F$ (shaded).")
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draw_surviving(ax)
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draw_ghosts(ax)
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draw_nodes(ax, pos, deg2, s=260, color=DEG2, edgecolors='black',
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linewidths=1.0, zorder=4)
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cx = sum(pos[('a', i)][0] for i in range(5)) / 5
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cy = sum(pos[('a', i)][1] for i in range(5)) / 5
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ax.text(cx, cy, '$F$', fontsize=16, ha='center', va='center',
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color='#a16207', zorder=5)
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ax.legend(handles=[
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Line2D([0], [0], marker='x', color=GHOST, lw=0, label='deleted (was $\\partial F_v$)'),
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Line2D([0], [0], marker='o', color='w', markerfacecolor=DEG2,
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markeredgecolor='black', label='degree-2 vertex'),
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], loc='upper left', fontsize=10)
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fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step1.png'),
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dpi=170, bbox_inches='tight'); plt.close(fig)
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# ----- Step 2: order the five degree-2 vertices clockwise as A_0..A_4 -----
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fig, ax = base_canvas(
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"Step 2: list the degree-2 vertices clockwise around $F$ as "
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"$A_0,\\dots,A_4$.")
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draw_surviving(ax)
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draw_nodes(ax, pos, deg2, s=300, color=DEG2, edgecolors='black',
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linewidths=1.0, zorder=4)
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for k, v in enumerate(A):
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x, y = pos[v]
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ax.annotate(f'$A_{k}$', (x, y), textcoords='offset points',
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xytext=(0, 0), ha='center', va='center', fontsize=10,
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fontweight='bold', color='black', zorder=5)
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# outward label too
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ax.annotate(f'$A_{k}$', (x * 1.18, y * 1.18), ha='center', va='center',
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fontsize=12, color='#a16207', zorder=5)
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fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step2.png'),
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dpi=170, bbox_inches='tight'); plt.close(fig)
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# ----- Step 3: add v_n joined to A_i, A_{i+1}, A_{i+2} -----
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fig, ax = base_canvas(
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"Step 3: add a vertex $v_n$ joined to $A_i, A_{i+1}, A_{i+2}$ "
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"(here $i=0$).")
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draw_surviving(ax)
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draw_nodes(ax, pos, deg2, s=300, color=DEG2, edgecolors='black',
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linewidths=1.0, zorder=4)
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for k, v in enumerate(A):
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ax.annotate(f'$A_{k}$', (pos[v][0] * 1.18, pos[v][1] * 1.18),
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ha='center', va='center', fontsize=12, color='#a16207', zorder=5)
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for u in apex_nbrs:
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(x0, y0), (x1, y1) = npos[v_n], pos[u]
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ax.plot([x0, x1], [y0, y1], color=APEX, lw=2.4, zorder=5)
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draw_nodes(ax, npos, [v_n], s=320, color=APEX, marker='s',
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edgecolors='black', linewidths=1.0, zorder=6)
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ax.annotate('$v_n$', npos[v_n], textcoords='offset points', xytext=(0, 14),
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ha='center', fontsize=12, fontweight='bold', color=APEX, zorder=7)
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fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step3.png'),
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dpi=170, bbox_inches='tight'); plt.close(fig)
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# ----- Step 4: add chord A_{i+3} A_{i+4}; the reduced dual -----
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fig, ax = base_canvas(
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"Step 4: add the edge $A_{i+3} A_{i+4}$. The result $\\widehat{G}'_{v,i}$ "
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"is again cubic and planar.")
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draw_surviving(ax)
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draw_nodes(ax, pos, deg2, s=300, color=DEG2, edgecolors='black',
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linewidths=1.0, zorder=4)
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for k, v in enumerate(A):
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ax.annotate(f'$A_{k}$', (pos[v][0] * 1.18, pos[v][1] * 1.18),
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ha='center', va='center', fontsize=12, color='#a16207', zorder=5)
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for u in apex_nbrs:
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(x0, y0), (x1, y1) = npos[v_n], pos[u]
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ax.plot([x0, x1], [y0, y1], color=APEX, lw=2.4, zorder=5)
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draw_nodes(ax, npos, [v_n], s=320, color=APEX, marker='s',
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edgecolors='black', linewidths=1.0, zorder=6)
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ax.annotate('$v_n$', npos[v_n], textcoords='offset points', xytext=(0, 14),
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ha='center', fontsize=12, fontweight='bold', color=APEX, zorder=7)
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(x0, y0), (x1, y1) = pos[chord[0]], pos[chord[1]]
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ax.plot([x0, x1], [y0, y1], color=CHORD, lw=2.8, zorder=5)
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fig.savefig(os.path.join(OUT_DIR, 'fig_reduced_dual_step4.png'),
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dpi=170, bbox_inches='tight'); plt.close(fig)
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print("wrote fig_reduced_dual_step1..4.png to", OUT_DIR)
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if __name__ == '__main__':
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main()
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"""Reduced dual: construction and verification.
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Test input is the icosahedron G (the unique 5-regular triangulation, n=12).
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Its dual G' is the dodecahedron (a cubic plane graph, 20 vertices). We pick a
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degree-5 vertex v of G -- equivalently a pentagonal face F_v of G' -- and apply
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the reduced-dual construction of Definition 2.1:
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1. delete the 5 dual vertices on the boundary of F_v (and incident edges),
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leaving 5 degree-2 vertices on a new face F;
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2. order those 5 vertices clockwise around F as A_0..A_4;
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3. add a vertex v_n joined to A_i, A_{i+1}, A_{i+2};
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4. add an edge A_{i+3} A_{i+4}.
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We verify the result is again a cubic plane graph, and report the triangulation
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it is the dual of (its face count = the primal vertex count), to see how the
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vertex count changes relative to n.
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The dodecahedron is built directly in its concentric "Schlegel" layout with
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F_v the inner pentagon, so the figures (draw_reduced_dual_steps.py) are clean.
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"""
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import math
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import networkx as nx
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# ---------------------------------------------------------------------------
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# Build G' = dodecahedron with concentric positions; F_v = inner pentagon.
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# Vertex families a (inner pentagon), b, c, d (outer pentagon), 5 each.
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# Angles increase *clockwise* (90 - 72*i deg) so index order is clockwise.
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# ---------------------------------------------------------------------------
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def build_dual():
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pos = {}
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R = {'a': 1.0, 'b': 2.2, 'c': 3.6, 'd': 4.8}
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for i in range(5):
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for fam in ('a', 'b'):
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th = math.radians(90 - 72 * i)
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pos[(fam, i)] = (R[fam] * math.cos(th), R[fam] * math.sin(th))
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for fam in ('c', 'd'):
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th = math.radians(90 - 72 * i - 36) # offset half a step
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pos[(fam, i)] = (R[fam] * math.cos(th), R[fam] * math.sin(th))
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Gp = nx.Graph()
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Gp.add_nodes_from(pos)
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for i in range(5):
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Gp.add_edge(('a', i), ('a', (i + 1) % 5)) # inner pentagon
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Gp.add_edge(('a', i), ('b', i)) # spokes a-b
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Gp.add_edge(('b', i), ('c', i)) # b-c
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Gp.add_edge(('b', i), ('c', (i - 1) % 5)) # b-c (other side)
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Gp.add_edge(('c', i), ('d', i)) # spokes c-d
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Gp.add_edge(('d', i), ('d', (i + 1) % 5)) # outer pentagon
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Fv_boundary = [('a', i) for i in range(5)] # inner pentagon
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return Gp, pos, Fv_boundary
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# ---------------------------------------------------------------------------
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# Face / dual helpers.
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# ---------------------------------------------------------------------------
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def faces_of(G):
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"""Return the list of faces (each a list of vertices) of a plane graph."""
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ok, emb = nx.check_planarity(G)
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assert ok, "graph is not planar"
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seen, faces = set(), []
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for u in emb:
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for v in emb[u]:
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if (u, v) not in seen:
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faces.append(emb.traverse_face(u, v, mark_half_edges=seen))
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return faces
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def dual_of(G):
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"""Combinatorial dual (all faces, including outer) of a plane graph."""
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faces = faces_of(G)
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edge_faces = {}
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for fi, face in enumerate(faces):
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for j in range(len(face)):
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e = frozenset((face[j], face[(j + 1) % len(face)]))
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edge_faces.setdefault(e, []).append(fi)
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D = nx.MultiGraph()
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D.add_nodes_from(range(len(faces)))
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for e, fs in edge_faces.items():
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if len(fs) == 2:
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D.add_edge(fs[0], fs[1])
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elif len(fs) == 1: # shouldn't happen for 2-connected G
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pass
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return D, faces
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# ---------------------------------------------------------------------------
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# The reduced-dual construction.
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# ---------------------------------------------------------------------------
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def clockwise_order(verts, pos):
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"""Order verts clockwise around their centroid, starting from the topmost."""
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cx = sum(pos[v][0] for v in verts) / len(verts)
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cy = sum(pos[v][1] for v in verts) / len(verts)
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ang = {v: math.atan2(pos[v][1] - cy, pos[v][0] - cx) for v in verts}
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ccw = sorted(verts, key=lambda v: ang[v]) # counterclockwise
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cw = list(reversed(ccw)) # clockwise
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start = max(range(len(cw)), key=lambda k: pos[cw[k]][1]) # topmost first
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return cw[start:] + cw[:start]
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def apply_reduction(Gp, pos, Fv_boundary, i=0):
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"""Apply Definition 2.1 and return a dict capturing each stage."""
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Ghat = Gp.copy()
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npos = dict(pos)
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# (1) delete the 5 boundary dual vertices of F_v
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Ghat.remove_nodes_from(Fv_boundary)
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deg2 = [v for v in Ghat if Ghat.degree(v) == 2]
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assert len(deg2) == 5, f"expected 5 degree-2 vertices, got {len(deg2)}"
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# (2) order them clockwise around the new face F
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A = clockwise_order(deg2, pos)
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# (3) new vertex v_n joined to A_i, A_{i+1}, A_{i+2}
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apex_nbrs = [A[(i + k) % 5] for k in range(3)]
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ax = sum(npos[v][0] for v in apex_nbrs) / 3
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ay = sum(npos[v][1] for v in apex_nbrs) / 3
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v_n = 'v_n'
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npos[v_n] = (ax * 0.55, ay * 0.55) # pull toward the 3 nbrs
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Ghat.add_node(v_n)
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for u in apex_nbrs:
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Ghat.add_edge(v_n, u)
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# (4) chord between the remaining two
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chord = (A[(i + 3) % 5], A[(i + 4) % 5])
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Ghat.add_edge(*chord)
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return {
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'Ghat': Ghat, 'pos': npos, 'A': A, 'v_n': v_n,
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'apex_nbrs': apex_nbrs, 'chord': chord,
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'deleted': list(Fv_boundary),
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}
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def main():
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Gp, pos, Fv = build_dual()
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# --- verify G' is the dodecahedron = dual of the icosahedron ---
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assert nx.check_planarity(Gp)[0]
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assert all(d == 3 for _, d in Gp.degree()), "G' not cubic"
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assert nx.is_isomorphic(Gp, nx.dodecahedral_graph()), "G' is not dodecahedron"
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Dico, _ = dual_of(Gp)
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Dico = nx.Graph(Dico)
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print(f"G (icosahedron) : dual of G' has {Dico.number_of_nodes()} vertices, "
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f"degrees {sorted({d for _, d in Dico.degree()})}")
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print(f"G' (dodecahedron): {Gp.number_of_nodes()} vertices, "
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f"{Gp.number_of_edges()} edges, "
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f"{len(faces_of(Gp))} faces; cubic={all(d==3 for _,d in Gp.degree())}")
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# --- apply the reduced-dual construction ---
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res = apply_reduction(Gp, pos, Fv, i=0)
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Ghat = res['Ghat']
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cubic = all(d == 3 for _, d in Ghat.degree())
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planar = nx.check_planarity(Ghat)[0]
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ghat_simple = (nx.number_of_selfloops(Ghat) == 0) # Graph: no parallels
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nfaces = len(faces_of(Ghat))
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print()
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print(f"reduced dual G^_v,i : {Ghat.number_of_nodes()} vertices, "
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f"{Ghat.number_of_edges()} edges, {nfaces} faces")
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print(f" cubic : {cubic}")
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print(f" planar : {planar}")
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print(f" simple : {ghat_simple}")
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# --- the triangulation it is dual to ---
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Dred_multi, _ = dual_of(Ghat)
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Dred = nx.Graph(Dred_multi)
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dred_simple = (Dred.number_of_edges() == Dred_multi.number_of_edges())
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is_tri = all(len(f) == 3 for f in faces_of(Dred)) if planar else None
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print()
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print(f"dual of reduced dual : {Dred.number_of_nodes()} vertices "
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f"(= faces of G^), degree seq "
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f"{sorted((d for _, d in Dred.degree()), reverse=True)}")
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print(f" is a triangulation : {is_tri}")
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print(f" simple : {dred_simple}")
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n = Dico.number_of_nodes()
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print()
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print(f"VERTEX COUNT: G has n = {n}; reduced triangulation has "
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f"{Dred.number_of_nodes()} (change = "
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||||
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@@ -115,4 +115,63 @@ $G$, a contradiction.
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Hence $\delta(G) \ge 5$.
|
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\end{proof}
|
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|
||||
\section{The reduced dual}
|
||||
|
||||
Write $G'$ for the dual of $G$: since $G$ is a triangulation, $G'$ is a cubic
|
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plane graph in which each vertex of $G$ corresponds to a face of $G'$, each face
|
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of $G$ to a vertex of $G'$, and each edge to a dual edge. A vertex of $G$ of
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degree $k$ corresponds to a $k$-gonal face of $G'$.
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By Lemma~\ref{lem:mindeg}, $\delta(G) \ge 5$, and Euler's formula gives
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$\sum_{u \in V(G)}(6 - \deg u) = 12$, so $G$ has a vertex of degree exactly $5$
|
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(indeed at least twelve). Fix such a vertex $v$. Its dual face $F_v$ is a
|
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pentagon, bounded by the five dual vertices corresponding to the five faces of
|
||||
$G$ incident to $v$.
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||||
\begin{definition}[Reduced dual]
|
||||
\label{def:reduced-dual}
|
||||
Let $v$ be a degree-$5$ vertex of $G$ with pentagonal dual face $F_v$, and fix an
|
||||
index $i \in \{0,1,2,3,4\}$. The \emph{reduced dual} $\widehat{G}'_{v,i}$ is the
|
||||
plane graph obtained from $G'$ as follows.
|
||||
\begin{enumerate}
|
||||
\item Delete the five dual vertices on the boundary of $F_v$, together with all
|
||||
edges incident to them. Each deleted vertex is cubic, with two edges on
|
||||
$\partial F_v$ and one edge leaving $F_v$; deleting the five boundary
|
||||
vertices therefore removes the five external edges as well, dropping their
|
||||
five outer endpoints from degree $3$ to degree $2$. These five degree-$2$
|
||||
vertices lie on the boundary of a single face $F$ of the resulting graph.
|
||||
\item List the five degree-$2$ vertices in clockwise order around $F$ as
|
||||
$A = (A_0, A_1, A_2, A_3, A_4)$.
|
||||
\item Add a new vertex $v_n$ and join it to $A_i$, $A_{i+1}$, and $A_{i+2}$
|
||||
(indices mod $5$) by three new edges.
|
||||
\item Add a new edge between $A_{i+3}$ and $A_{i+4}$ (indices mod $5$).
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
Steps (3) and (4) restore cubicity: $A_i, A_{i+1}, A_{i+2}$ each gain one edge to
|
||||
$v_n$ and $A_{i+3}, A_{i+4}$ each gain the new edge, so all five return to degree
|
||||
$3$, and $v_n$ has degree $3$. Since $A_i,\dots,A_{i+2}$ and $A_{i+3}, A_{i+4}$
|
||||
are each consecutive along $\partial F$, the new vertex and edge can be drawn
|
||||
inside $F$ without crossings, so $\widehat{G}'_{v,i}$ is again a cubic plane
|
||||
graph. The construction depends on the choice of $i$ up to the rotational
|
||||
symmetry of $A$.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step1.png}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step2.png}\\[0.5em]
|
||||
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step3.png}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step4.png}
|
||||
\caption{The four steps of Definition~\ref{def:reduced-dual}, illustrated on
|
||||
$G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner
|
||||
pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$,
|
||||
leaving five degree-$2$ vertices on a new face $F$. Top right: order them
|
||||
clockwise as $A_0,\dots,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$.
|
||||
Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph
|
||||
$\widehat{G}'_{v,0}$.}
|
||||
\label{fig:reduced-dual-steps}
|
||||
\end{figure}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user