Revert "coloring_nested_tire_graphs: empirical uniqueness-break figure (HM_0)"
This reverts commit 22fa29a8bb.
This commit is contained in:
@@ -1,226 +0,0 @@
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"""Draw an empirical uniqueness-break case using TRUE planar
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embedding coordinates (not spring layout).
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We need a LOW-SIDE face f of H_d such that f's interior contains
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H_{d-1} edges from multiple H_{d-1} faces.
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The most reliable low-side face: the OUTER (unbounded) face of H_d.
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For d >= 2, the outer face of H_d contains all of H_{d-1} (= depth-
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(d-1) subgraph) plus pendants.
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Algorithm:
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- For HM_0 cut #1 side 1 (a known interesting case), use the
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primal planar embedding for vertex positions.
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- Identify the outer face of H_2 (= largest face by area, or by
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Sage's outer-face-of-embedding).
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- Verify it contains H_1 edges from multiple H_1 faces in its
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interior region.
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- Draw.
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"""
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import os, sys
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from collections import defaultdict
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import matplotlib.pyplot as plt
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from sage.all import Graph
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from cut_depth_label import (
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parse_planar_code, HM_FILE,
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apply_procedure, compute_nice_layout,
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)
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from cut_tire_tree import compute_H_d_faces
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from tree_structure_sweep import all_six_edge_cuts
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def planar_layout_for_H(H):
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"""Return planar layout of H using Sage's planar embedding."""
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# Try planar layout
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H2 = H.copy()
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try:
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H2.is_planar(set_embedding=True)
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# Use Sage's planar layout
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pos = H2.layout(layout='planar')
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# pos values are (x, y) but Sage returns dict v -> (x,y)
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return pos
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except Exception as e:
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print(f' Planar layout failed: {e}')
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return None
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def main():
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gs = parse_planar_code(HM_FILE)
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G = gs[0]
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G.is_planar(set_embedding=True)
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base_pos = compute_nice_layout(G)
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cuts = all_six_edge_cuts(G, max_cuts=5)
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S, cut_edges = cuts[1]
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S1 = frozenset(G.vertices()) - S
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H, _, ed, _, _, _ = apply_procedure(
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G, S1, cut_edges, base_pos, '1',
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pendant_start_id=max(G.vertices()) + 501)
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print(f'|V(H)|={H.order()}, |E(H)|={H.size()}')
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print(f'Depths: {sorted(set(ed.values()))}')
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# Use Sage's planar layout for H
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pos = planar_layout_for_H(H)
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if pos is None:
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pos = H.layout(layout='spring')
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print(f'Got positions for {len(pos)} vertices')
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# Compute faces of H_1, H_2, H_3
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faces_1, _ = compute_H_d_faces(H, ed, 1)
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faces_2, _ = compute_H_d_faces(H, ed, 2)
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faces_3, _ = compute_H_d_faces(H, ed, 3)
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print(f'H_1: {len(faces_1)} faces, lengths {[len(f) for f in faces_1]}')
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print(f'H_2: {len(faces_2)} faces, lengths {[len(f) for f in faces_2]}')
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print(f'H_3: {len(faces_3)} faces, lengths {[len(f) for f in faces_3]}')
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# Identify the LOW-side face of H_2.
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# A low-side face contains depth-<2 edges (= pendants + H_1 edges).
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# Equivalently: it's the face whose interior in the planar embedding
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# contains the pendant vertices.
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pendant_verts = {v for e, d in ed.items() if d == 0 for v in e}
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# We pick the face whose boundary walk encloses the pendants.
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# In Sage's planar embedding, this is the "outer" face often.
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# Heuristic: the H_2 face NOT containing H_3 edges in its interior.
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# Or: the largest face by vertex count.
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# Most reliable: the face that, when removed from the plane, the
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# pendants are in the remaining unbounded region.
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# Use Sage's faces() — the first face is the outer face by default.
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# For now, pick the face with the most vertices.
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target_face_idx = max(range(len(faces_2)),
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key=lambda i: len(set(v for u, v in faces_2[i])
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| set(u for u, v in faces_2[i])))
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target_face = faces_2[target_face_idx]
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target_verts = set()
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for u, v in target_face:
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target_verts.add(u); target_verts.add(v)
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print(f'\nTarget face (H_2 face {target_face_idx}): {len(target_face)} '
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f'edges, {len(target_verts)} verts')
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print(f' vertices: {sorted(target_verts)}')
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# Identify H_1 edges in the "interior" of the target face.
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# Heuristic: H_1 edges whose endpoints are NOT on the target face
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# boundary, OR endpoints on the boundary but the edge "points
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# inward" by planar embedding.
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# For simplicity, look at all H_1 edges with at least one endpoint
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# on target_verts and the OTHER endpoint NOT on H_2's outer
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# boundary.
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h1_in_interior = []
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h1_face_of_edge = {}
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for fi, walk in enumerate(faces_1):
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for u, v in walk:
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e = tuple(sorted((u, v)))
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h1_face_of_edge.setdefault(e, []).append(fi)
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for e, d in ed.items():
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if d != 1:
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continue
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# An H_1 edge "in target face's interior" if at least one
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# endpoint is in target_verts (= boundary of target face).
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if e[0] in target_verts or e[1] in target_verts:
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h1_in_interior.append(e)
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h1_in_int_by_face = defaultdict(list)
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for e in h1_in_interior:
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for fi in h1_face_of_edge.get(e, []):
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h1_in_int_by_face[fi].append(e)
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print(f'\nH_1 edges adjacent to target face boundary:')
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for fi, eds in sorted(h1_in_int_by_face.items()):
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print(f' H_1 face {fi}: {len(set(eds))} edges')
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# Draw
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fig, ax = plt.subplots(figsize=(11, 10))
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# All H edges in light gray as background
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for u, v in H.edges(labels=False):
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e = tuple(sorted((u, v)))
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x1, y1 = pos[u]; x2, y2 = pos[v]
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de = ed.get(e, -1)
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if de == 0:
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color = '#888888'; lw = 0.8; alpha = 0.4
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elif de == 1:
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color = '#4477CC'; lw = 2.0; alpha = 0.8
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elif de == 2:
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color = '#DD7733'; lw = 3.5; alpha = 1.0
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elif de == 3:
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color = '#999999'; lw = 0.8; alpha = 0.3
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else:
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color = '#CCCCCC'; lw = 0.5; alpha = 0.2
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ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw,
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alpha=alpha, zorder=2)
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# Highlight target H_2 face boundary
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for u, v in target_face:
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x1, y1 = pos[u]; x2, y2 = pos[v]
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ax.plot([x1, x2], [y1, y2], color='#DD7733', linewidth=5.5,
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alpha=1.0, zorder=4, solid_capstyle='round')
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# H_1 edges colored by their H_1 face
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h1_colors = {'face 0': '#22AA88', 'face 1': '#AA22AA',
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'face 2': '#225599'}
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color_list = ['#22AA88', '#AA22AA', '#225599']
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for fi, eds in sorted(h1_in_int_by_face.items()):
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col = color_list[fi % len(color_list)]
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for e in set(eds):
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u, v = e
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x1, y1 = pos[u]; x2, y2 = pos[v]
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ax.plot([x1, x2], [y1, y2], color=col, linewidth=4.0,
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alpha=0.95, zorder=5, solid_capstyle='round')
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# Vertices
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for v in H.vertices():
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x, y = pos[v]
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in_target = v in target_verts
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is_pendant = v in pendant_verts
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if is_pendant:
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ax.plot(x, y, 'o', color='#CC3333', markersize=6, zorder=6)
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elif in_target:
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ax.plot(x, y, 'o', color='#DD7733', markersize=7, zorder=6,
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markeredgecolor='black', markeredgewidth=1)
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ax.annotate(str(v), (x, y), textcoords='offset points',
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xytext=(7, 7), fontsize=9, color='black', zorder=7)
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else:
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ax.plot(x, y, 'ko', markersize=4, zorder=6)
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from matplotlib.lines import Line2D
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legend = [
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Line2D([0], [0], color='#DD7733', lw=5,
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label=f'$H_2$ face {target_face_idx} boundary '
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f'({len(target_face)} edges)'),
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Line2D([0], [0], color='#22AA88', lw=4,
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label=f'$H_1$ edges in face 0'),
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Line2D([0], [0], color='#AA22AA', lw=4,
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label=f'$H_1$ edges in face 1'),
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Line2D([0], [0], color='#225599', lw=4,
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label=f'$H_1$ edges in face 2'),
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Line2D([0], [0], color='#4477CC', lw=1.5, label='other $H_1$ edges'),
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Line2D([0], [0], color='#888888', lw=0.8, label='pendants ($d=0$)'),
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Line2D([0], [0], color='#999999', lw=0.8, label='$H_3+$ edges'),
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Line2D([0], [0], marker='o', color='#CC3333', lw=0,
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label='pendant vertex', markersize=8),
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Line2D([0], [0], marker='o', color='#DD7733', lw=0,
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label='$H_2$ face vertex', markersize=8,
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markeredgecolor='black'),
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]
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ax.legend(handles=legend, loc='upper left', fontsize=8,
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framealpha=0.95)
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ax.set_aspect('equal')
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ax.axis('off')
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n_h1_faces = len(h1_in_int_by_face)
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ax.set_title(
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f'HM_0 cut #1 side 1, $d=2$: this $H_2$ face has $H_1$ edges '
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f'from {n_h1_faces} different $H_1$ faces adjacent\n'
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f'(planar embedding from Sage; $H_2$ face shown in orange; '
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f'$H_1$ edges grouped by face)')
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out = os.path.join(os.path.dirname(HERE), 'notes',
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'uniqueness_break_example.pdf')
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fig.savefig(out, bbox_inches='tight', dpi=120)
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print(f'\nWrote {out}')
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if __name__ == '__main__':
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main()
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@@ -1,171 +0,0 @@
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"""Find an empirical case where the low-side face of H_d spans
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multiple faces of H_{d-1} (= the case high-side restriction is
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needed for).
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For each (G, cut, side), iterate (d, face) and check:
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- Is `face` the low-side face of H_d?
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- Does face's interior contain H_{d-1} edges from multiple
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H_{d-1} faces?
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Output: a clear description of the case + edges of H_d, H_{d-1},
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and their face structures, so we can draw it.
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"""
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import os, sys
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from collections import defaultdict
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from sage.all import Graph, graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from cut_depth_label import (
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parse_planar_code, HM_FILE,
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apply_procedure, compute_nice_layout,
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)
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from cut_tire_tree import build_tree, compute_H_d_faces
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from tree_structure_sweep import all_six_edge_cuts
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def classify_face(face, edge_depth, d):
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"""Returns 'low', 'high', or 'mixed' based on adjacency to non-Hd
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edges by depth around the face vertices."""
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verts = set()
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for u, v in face:
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verts.add(u); verts.add(v)
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seen_lower = False
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seen_higher = False
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# For each vertex in face, look at incident edges NOT on this face
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face_e_set = {tuple(sorted(e)) for e in face}
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# Approach: any incident edge with depth < d → "lower" hint;
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# any with depth > d → "higher" hint. (The face's interior could
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# contain stuff via vertex-adjacency.)
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return None # not used, we'll classify differently
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def find_face_of_edge(faces, e_norm):
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"""Find which face(s) of a graph have edge e_norm in their walk."""
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matches = []
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for fi, walk in enumerate(faces):
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for u, v in walk:
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if tuple(sorted((u, v))) == e_norm:
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matches.append(fi)
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break
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return matches
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def analyze_cut(G, cut_idx, cuts, base_pos):
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S, cut_edges = cuts[cut_idx]
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S0, S1 = S, frozenset(G.vertices()) - S
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for side, S_side in [('0', S0), ('1', S1)]:
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if len(S_side) < 6 or len(S_side) > G.order() - 6:
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continue
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try:
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H, _, ed, _, _, _ = apply_procedure(
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G, S_side, cut_edges, base_pos, side,
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pendant_start_id=max(G.vertices()) + 1 + (
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0 if side == '0' else 500))
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except Exception:
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continue
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if not ed:
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continue
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max_d = max(ed.values())
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for d in range(2, max_d + 1):
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faces_d, H_d = compute_H_d_faces(H, ed, d)
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faces_dm1, H_dm1 = compute_H_d_faces(H, ed, d - 1)
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if not faces_d or not faces_dm1 or len(faces_dm1) < 2:
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continue
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# For each face of H_d, find which H_{d-1} edges are
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# "inside" it (= H_{d-1} edges with both endpoints among
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# face's vertex closure, intuitively).
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# Simpler: a low-side face of H_d is one whose interior
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# contains depth-<d edges. Detect by adjacency:
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# any vertex of face has incident depth-<d edges → low.
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for fi, face in enumerate(faces_d):
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face_verts = set()
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for u, v in face:
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face_verts.add(u); face_verts.add(v)
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# Edges of H_{d-1} adjacent to face_verts (= edges of
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# depth d-1 incident to a face vertex)
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hdm1_neighbor_edges = set()
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for v in face_verts:
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for nb in H.neighbors(v):
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e = tuple(sorted((v, nb)))
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if ed.get(e) == d - 1:
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hdm1_neighbor_edges.add(e)
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if len(hdm1_neighbor_edges) < 2:
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continue
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# Which H_{d-1} faces do these edges belong to?
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# Edges in H_{d-1} face boundary
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hdm1_face_of_edge = {}
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for fjk, walk in enumerate(faces_dm1):
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for u, v in walk:
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e = tuple(sorted((u, v)))
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hdm1_face_of_edge.setdefault(e, set()).add(fjk)
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faces_touched = set()
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edge_to_face = defaultdict(list)
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for e in hdm1_neighbor_edges:
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for fjk in hdm1_face_of_edge.get(e, set()):
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faces_touched.add(fjk)
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edge_to_face[fjk].append(e)
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# Filter: must have unique edges across faces (the
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# same edge in multiple faces is the trivial shared-
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# boundary case, not interesting).
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edge_unique_to_face = defaultdict(set)
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for fjk, eds in edge_to_face.items():
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for e in eds:
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edge_unique_to_face[e].add(fjk)
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# Count edges that appear in only one face
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unique_edges = {e for e, fjks in edge_unique_to_face.items()
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if len(fjks) == 1}
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if len(faces_touched) >= 2 and len(face) >= 6 and \
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len(unique_edges) >= 4:
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print(f'\n=== FOUND: side {side}, d={d}, face {fi} of '
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f'H_d spans {len(faces_touched)} H_{{d-1}} '
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f'faces ===', flush=True)
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print(f' Cut: |S0|={len(S0)}, |S1|={len(S1)}')
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print(f' H_d face boundary edges ({len(face)}):',
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flush=True)
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for u, v in face[:8]:
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print(f' ({u},{v})')
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if len(face) > 8:
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print(f' ... ({len(face)-8} more)')
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print(f' H_{{d-1}} edges adjacent to face '
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f'(broken across H_{{d-1}} faces):')
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for fjk in sorted(faces_touched):
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print(f' H_{{d-1}} face {fjk}: '
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f'{[(min(e),max(e)) for e in edge_to_face[fjk][:5]]}')
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return {
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'side': side, 'd': d, 'face_idx': fi,
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'face_walk': face, 'face_verts': face_verts,
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'H_d_edges': [(u,v) for u,v in face],
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'H_dm1_faces': {fjk: edge_to_face[fjk]
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for fjk in faces_touched},
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'all_H_d_faces': faces_d,
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'all_H_dm1_faces': faces_dm1,
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'edge_depth': ed,
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}
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return None
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def main():
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gs = parse_planar_code(HM_FILE)
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candidates = [('Dodecahedron', graphs.DodecahedralGraph())]
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for i, g in enumerate(gs):
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candidates.append((f'HM_{i}', g))
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for name, G in candidates:
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G.is_planar(set_embedding=True)
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base_pos = compute_nice_layout(G)
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cuts = all_six_edge_cuts(G, max_cuts=30)
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print(f'\n=== {name}: testing {len(cuts)} cuts ===', flush=True)
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for ci in range(len(cuts)):
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res = analyze_cut(G, ci, cuts, base_pos)
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if res:
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print(f'\nFOUND case in cut #{ci} of {name}', flush=True)
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res['graph_name'] = name
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res['cut_idx'] = ci
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return res
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print('No example found.')
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return None
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Binary file not shown.
@@ -85,37 +85,6 @@ By contrast, face $A$ (high-side, inside the inner cycle) sits
|
||||
entirely inside face $X$ of $H_{d-1}$. Unique parent. This is
|
||||
why the forest proposition restricts to high-side faces.
|
||||
|
||||
\paragraph{An empirically observed case.} Holton--McKay graph
|
||||
$\mathrm{HM}_0$, $6$-edge cut \#$1$, side $1$ (with $|S_1| = 28$).
|
||||
At $d = 2$: $H_2$ has three faces of lengths $4, 4, 12$; $H_1$ also
|
||||
has three faces of lengths $4, 4, 12$. The outer face of $H_2$
|
||||
(the length-$12$ one, $7$ vertices: $\{16,19,20,21,23,24,28\}$) is
|
||||
low-side --- in its interior live the depth-$0$ pendants and all
|
||||
of the depth-$1$ ($H_1$) edges. Those $H_1$ edges are spread
|
||||
across all three faces of $H_1$:
|
||||
\begin{itemize}
|
||||
\item $H_1$ face $0$: contributes edge $(15,19)$.
|
||||
\item $H_1$ face $1$: contributes edge $(17,21)$.
|
||||
\item $H_1$ face $2$: contributes edges $(23, 27), (28, 33), (24, 29),
|
||||
(28, 34)$.
|
||||
\end{itemize}
|
||||
So this single low-side face of $H_2$ has $H_1$ edges from
|
||||
\emph{three} different $H_1$ faces adjacent to its boundary --- no
|
||||
single $H_1$ face contains all of them, hence no unique parent.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.95\textwidth]{uniqueness_break_example.pdf}
|
||||
\end{center}
|
||||
|
||||
The orange ring is the boundary of the low-side $H_2$ face. The
|
||||
three coloured edges (green, purple, blue) are $H_1$ edges
|
||||
adjacent to the orange ring's vertices, coloured by which $H_1$
|
||||
face they belong to. The diagram is laid out using Sage's planar
|
||||
embedding of $H$. If we tried to assign this $H_2$ face a single
|
||||
``parent'' in $H_1$, none of the three $H_1$ faces would contain
|
||||
it. $T_\partial$ exists precisely to handle this low-side
|
||||
exception.
|
||||
|
||||
\paragraph{The coverage gap.} Empirically
|
||||
(\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side
|
||||
$0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a
|
||||
|
||||
Binary file not shown.
Reference in New Issue
Block a user