didericis
7183dc1b67
Defines level cycles, edge switches, surface switches, and facial depth on level components of plane triangulations. Proves outerplanarity of level components and a depth-descent lemma. Introduces balanced surface switches and proves they remove a depth-d level cycle while creating 1-2 new depth-(d-1) cycles. Documents the 9-vertex counterexample where no balanced switch exists and sketches preprocessing toward balancedness, leaving general termination open. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
119 lines
4.3 KiB
Python
119 lines
4.3 KiB
Python
"""Demonstrate the preprocessing strategy on the 9-vertex example.
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Start: F = (0,3,6) at depth 1, no balanced surface switch exists
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(F has no edge of "span 1" -- no edge with a single outer-cycle
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vertex between the endpoints, hence no ear neighbour).
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Step: perform the (unbalanced) surface switch on edge uv = 03, with
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F' = (0,2,3) and third vertex x = 2; in Case (ii) the flip removes 03
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and adds wx = 62.
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Result: A = (0,2,6) at depth 1 has edge 02 at span 1, so the ear
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(0,1,2) is now a balanced-switch target.
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"""
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import os
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import math
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import networkx as nx
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
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n = 9
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POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
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math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
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OUTER_EDGES = [(i, (i + 1) % n) for i in range(n)]
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outer_set = {frozenset(e) for e in OUTER_EDGES}
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def face_edges(f):
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return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
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frozenset((f[0], f[2]))}
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def compute_depths(faces):
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D = nx.Graph()
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D.add_nodes_from(range(len(faces)))
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for i, fi in enumerate(faces):
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for j, fj in enumerate(faces):
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if i < j and face_edges(fi) & face_edges(fj):
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D.add_edge(i, j)
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B = [i for i, f in enumerate(faces)
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if len(face_edges(f) & outer_set) >= 1]
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return {i: min(nx.shortest_path_length(D, i, b) for b in B)
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for i in range(len(faces))}
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CHORDS_BEFORE = [(0, 2), (0, 3), (3, 5), (3, 6), (0, 6), (6, 8)]
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FACES_BEFORE = [
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(0, 1, 2), (0, 2, 3), (3, 4, 5), (3, 5, 6),
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(6, 7, 8), (6, 8, 0), (0, 3, 6),
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]
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# After non-balanced switch on edge 03: remove 03, add 26
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CHORDS_AFTER = [c for c in CHORDS_BEFORE if set(c) != {0, 3}] + [(2, 6)]
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FACES_AFTER = [
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(0, 1, 2), (3, 4, 5), (3, 5, 6),
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(6, 7, 8), (6, 8, 0),
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(0, 2, 6), (2, 3, 6),
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]
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def draw(ax, faces, chords, depth, title, highlight_edges=None,
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green_edges=None):
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palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
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edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
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for i, f in enumerate(faces):
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d = depth[i]
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poly = Polygon([POS[v] for v in f], closed=True,
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facecolor=palette.get(d, '#ddd'),
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edgecolor=edge_pal.get(d, '#333'),
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linewidth=1.4, alpha=0.7, zorder=0)
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ax.add_patch(poly)
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cx = sum(POS[v][0] for v in f) / 3
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cy = sum(POS[v][1] for v in f) / 3
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ax.text(cx, cy, str(d), ha='center', va='center', fontsize=11,
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color=edge_pal.get(d, '#333'), fontweight='bold')
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for (a, b) in OUTER_EDGES + chords:
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color = '#333'; lw = 1.2
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if highlight_edges and ((a, b) in highlight_edges or
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(b, a) in highlight_edges):
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color = '#dc2626'; lw = 3.0
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if green_edges and ((a, b) in green_edges or
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(b, a) in green_edges):
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color = '#16a34a'; lw = 3.0
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ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
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color=color, linewidth=lw, zorder=1)
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for i, (x, y) in POS.items():
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ax.scatter([x], [y], s=270, c='#1f2937', edgecolors='black',
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linewidths=1.0, zorder=2)
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ax.text(x, y, str(i), ha='center', va='center',
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fontsize=9, color='white', fontweight='bold', zorder=3)
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ax.set_aspect('equal'); ax.axis('off')
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ax.set_xlim(-1.3, 1.3); ax.set_ylim(-1.3, 1.3)
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ax.set_title(title, fontsize=11)
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depth_before = compute_depths(FACES_BEFORE)
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depth_after = compute_depths(FACES_AFTER)
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print('BEFORE:')
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for i, f in enumerate(FACES_BEFORE):
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print(f' {f} -> depth {depth_before[i]}')
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print('AFTER:')
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for i, f in enumerate(FACES_AFTER):
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print(f' {f} -> depth {depth_after[i]}')
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fig, axes = plt.subplots(1, 2, figsize=(14, 7))
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draw(axes[0], FACES_BEFORE, CHORDS_BEFORE, depth_before,
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'Before: F=(0,3,6) depth 1; spans (2,2,2) so no ear neighbour',
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highlight_edges=[(0, 3)])
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draw(axes[1], FACES_AFTER, CHORDS_AFTER, depth_after,
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'After non-balanced switch 03->26: A=(0,2,6) depth 1; edge 02 has span 1',
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green_edges=[(2, 6)], highlight_edges=[(0, 2)])
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_preprocessing.png')
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fig.savefig(out, dpi=180, bbox_inches='tight')
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plt.close(fig)
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print(f'wrote {out}')
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