dual_decomposition: split iterated reduction into companion paper
- Move the iterated-reduction algorithm, its two structural lemmas
(exactly-one-match, all-distinct-exists), and the n=14 trace figure
into a new companion paper at
papers/dual_decomposition_iterated_reduction/. Figures and figure
scripts moved via git mv (history preserved).
- In the main paper, Section 3 ("An iterated reduction") becomes
Section 3 "Cubic-graph edge contraction" (just the contraction
definition + 4-face theorem).
- Restructure Section 4 to host both the original face-monochromatic-pair
conjecture (clauses 1-3) and its strengthening (adds clause 4) as
separate conjectures, after briefly experimenting with folding them
into one. The empirical evidence is asymmetric (n<=21 for (1)-(3),
n<=18 for the full set), which the two-conjecture split presents more
honestly. The companion-paper reference is now in Section 4's intro.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
-219
@@ -1,219 +0,0 @@
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"""Draw the iterated reduction algorithm's trace on the dodecahedron.
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Produces three figures:
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fig_alg_step0.png -- G' (dodecahedron) with F_v (inner pentagon) shaded.
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fig_alg_step1.png -- H_1 (post step 1), 3-edge-coloured; 4 protected edges.
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fig_alg_step2.png -- H_2 (post step 2), 3-edge-coloured; 8 protected edges;
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algorithm terminates.
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Run with: sage experiments/draw_iterated_reduction.py
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"""
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from sage.all import Graph
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from sage.graphs.graph_coloring import edge_coloring
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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import math
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import os
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OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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C = ['#dc2626', '#16a34a', '#2563eb'] # proper-edge-colour palette
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GRAY = '#9ca3af'
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DARK = '#374151'
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HIGHLIGHT = '#fef3c7'
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def dodecahedron_positions():
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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)
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pos[(fam, i)] = (R[fam] * math.cos(th), R[fam] * math.sin(th))
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return pos
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def build_dodecahedron():
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edges = []
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for i in range(5):
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edges.append((('a', i), ('a', (i + 1) % 5)))
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edges.append((('a', i), ('b', i)))
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edges.append((('b', i), ('c', i)))
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edges.append((('b', i), ('c', (i - 1) % 5)))
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edges.append((('c', i), ('d', i)))
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edges.append((('d', i), ('d', (i + 1) % 5)))
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G = Graph(edges, multiedges=False, loops=False)
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G.is_planar(set_embedding=True)
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return G
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def find_safe_pentagonal_face(G, protected):
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for face in G.faces():
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if len(face) != 5:
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continue
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boundary = [u for (u, v) in face]
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boundary_edges = [frozenset([u, v]) for (u, v) in face]
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externals = []
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A = []
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for B_k in boundary:
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outer = [w for w in G.neighbor_iterator(B_k) if w not in boundary]
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if len(outer) != 1:
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break
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externals.append(frozenset([B_k, outer[0]]))
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A.append(outer[0])
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else:
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if not any(e in protected for e in boundary_edges + externals):
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return boundary, externals, A
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return None
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def valid_indices(f_vec):
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out = []
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for i in range(5):
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if f_vec[(i + 3) % 5] != f_vec[(i + 4) % 5]:
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continue
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if len({f_vec[i], f_vec[(i + 1) % 5], f_vec[(i + 2) % 5]}) == 3:
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out.append(i)
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return out
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def draw(ax, G, pos, *, coloring=None, protected=None,
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shade_face=None):
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if shade_face:
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poly = [pos[v] for v in shade_face]
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ax.add_patch(Polygon(poly, closed=True, facecolor=HIGHLIGHT,
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edgecolor='none', zorder=0))
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protected = protected or set()
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for u, v in G.edges(labels=False):
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e = frozenset([u, v])
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c = C[coloring[e]] if coloring is not None else GRAY
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lw = 3.8 if e in protected else 1.4
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(x0, y0), (x1, y1) = pos[u], pos[v]
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ax.plot([x0, x1], [y0, y1], color=c, lw=lw, zorder=2)
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for v in G.vertices(sort=False):
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x, y = pos[v]
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if isinstance(v, tuple) and v[0] == 'v_n':
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t = v[1]
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ax.scatter(x, y, s=320, color=HIGHLIGHT, marker='s',
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edgecolors='black', linewidths=1.2, zorder=4)
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ax.annotate(f'$v_n^{{({t})}}$', (x, y),
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textcoords='offset points', xytext=(16, 16),
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ha='left', fontsize=14, fontweight='bold',
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color=DARK, zorder=6,
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bbox=dict(boxstyle='round,pad=0.2', fc='white',
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ec=DARK, lw=0.6))
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else:
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ax.scatter(x, y, s=70, color=DARK, zorder=3)
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ax.set_aspect('equal')
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ax.axis('off')
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def main():
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G = build_dodecahedron()
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pos = dodecahedron_positions()
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F_v = [('a', i) for i in range(5)]
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# ----- Step 0: G' with F_v shaded -----
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fig, ax = plt.subplots(figsize=(8, 8))
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draw(ax, G, pos, shade_face=F_v)
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step0.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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# ----- Step 1: Definition 2.1 at F_v with i_1 = 0 -----
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safe = find_safe_pentagonal_face(G, set())
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boundary_1, externals_1, A_1 = safe
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G1 = G.copy()
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for v in boundary_1:
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G1.delete_vertex(v)
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v_n_1 = ('v_n', 1)
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G1.add_vertex(v_n_1)
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G1.add_edge(v_n_1, A_1[0])
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G1.add_edge(v_n_1, A_1[1])
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G1.add_edge(v_n_1, A_1[2])
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G1.add_edge(A_1[3], A_1[4])
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G1.is_planar(set_embedding=True)
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pos1 = {v: p for v, p in pos.items() if v not in boundary_1}
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cx = (pos[A_1[0]][0] + pos[A_1[1]][0] + pos[A_1[2]][0]) / 3
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cy = (pos[A_1[0]][1] + pos[A_1[1]][1] + pos[A_1[2]][1]) / 3
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pos1[v_n_1] = (cx * 0.55, cy * 0.55)
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cols = edge_coloring(G1, value_only=False)
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coloring = {}
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for k, edge_list in enumerate(cols):
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for u, v in edge_list:
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coloring[frozenset([u, v])] = k
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E = {
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frozenset([v_n_1, A_1[1]]), # spike
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frozenset([v_n_1, A_1[0]]), # side-0
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frozenset([v_n_1, A_1[2]]), # side-1
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frozenset([A_1[3], A_1[4]]), # merged
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}
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fig, ax = plt.subplots(figsize=(8, 8))
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draw(ax, G1, pos1, coloring=coloring, protected=E)
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step1.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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# ----- Step 2: reduce at the only remaining safe face (outer pentagon) -----
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safe = find_safe_pentagonal_face(G1, E)
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if safe is None:
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print("ERROR: expected an outer pentagonal face but none found.")
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return
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boundary_2, externals_2, A_2 = safe
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f_vec = [coloring[e] for e in externals_2]
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choices = valid_indices(f_vec)
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if not choices:
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print(f"ERROR: f-vector {f_vec} has no valid index.")
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return
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i_t = choices[0]
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G2 = G1.copy()
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for v in boundary_2:
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G2.delete_vertex(v)
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v_n_2 = ('v_n', 2)
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G2.add_vertex(v_n_2)
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G2.add_edge(v_n_2, A_2[i_t])
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G2.add_edge(v_n_2, A_2[(i_t + 1) % 5])
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G2.add_edge(v_n_2, A_2[(i_t + 2) % 5])
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G2.add_edge(A_2[(i_t + 3) % 5], A_2[(i_t + 4) % 5])
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G2.is_planar(set_embedding=True)
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coloring2 = {e: c for e, c in coloring.items()
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if not any(u in boundary_2 for u in e)}
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side_0_2 = frozenset([v_n_2, A_2[i_t]])
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spike_2 = frozenset([v_n_2, A_2[(i_t + 1) % 5]])
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side_1_2 = frozenset([v_n_2, A_2[(i_t + 2) % 5]])
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merged_2 = frozenset([A_2[(i_t + 3) % 5], A_2[(i_t + 4) % 5]])
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coloring2[side_0_2] = coloring[externals_2[i_t]]
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coloring2[spike_2] = coloring[externals_2[(i_t + 1) % 5]]
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coloring2[side_1_2] = coloring[externals_2[(i_t + 2) % 5]]
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coloring2[merged_2] = coloring[externals_2[(i_t + 3) % 5]]
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pos2 = {v: p for v, p in pos1.items() if v not in boundary_2}
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nbrs = [A_2[i_t], A_2[(i_t + 1) % 5], A_2[(i_t + 2) % 5]]
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cx = sum(pos2[a][0] for a in nbrs) / 3
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cy = sum(pos2[a][1] for a in nbrs) / 3
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r = math.hypot(cx, cy)
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# v_n^{(2)} lies outside the surviving graph (the deleted d's were outermost)
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target_r = 5.0
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pos2[v_n_2] = (cx * target_r / r, cy * target_r / r)
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E |= {side_0_2, spike_2, side_1_2, merged_2}
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fig, ax = plt.subplots(figsize=(8, 8))
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draw(ax, G2, pos2, coloring=coloring2, protected=E)
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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print(f"Wrote fig_alg_step{{0,1,2}}.png to {OUT_DIR}")
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if __name__ == '__main__':
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main()
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-520
@@ -1,520 +0,0 @@
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"""Draw the iterated reduction trace on the smallest triangulation where
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the chord-apex + Kempe-cycle property is satisfied: the first min-degree-5
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plantri triangulation on n = 14 vertices, found by search_kempe_property.py.
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Overwrites fig_alg_step{0,1,2}.png in the paper directory with this
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triangulation's trace (replacing the dodecahedron version).
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Run with: sage experiments/draw_iterated_reduction_n14.py
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"""
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from sage.all import Graph
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from sage.graphs.graph_generators import graphs
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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import math
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import os
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def tutte_layout(G_sage, avoid_verts=None, iterations=300):
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"""Tutte's barycentric embedding: pick the largest face whose vertex set
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avoids `avoid_verts` as the outer face, place its vertices on a regular
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polygon, then iterate each interior vertex to the barycenter of its
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neighbors. For 3-connected planar graphs this converges to the unique
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straight-line planar embedding with the chosen outer face --- balanced
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by construction and free of edge crossings."""
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avoid = set(avoid_verts or ())
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candidates = []
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for face in G_sage.faces():
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verts = [u for (u, v) in face]
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if not (set(verts) & avoid):
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candidates.append(verts)
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if not candidates:
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outer = [u for (u, v) in max(G_sage.faces(), key=len)]
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else:
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outer = max(candidates, key=len)
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n_outer = len(outer)
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pos = {}
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for k, v in enumerate(outer):
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ang = 2 * math.pi * k / n_outer + math.pi / 2
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pos[v] = (math.cos(ang), math.sin(ang))
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interior = [v for v in G_sage.vertex_iterator() if v not in pos]
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for v in interior:
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pos[v] = (0.0, 0.0)
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for _ in range(iterations):
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new_pos = dict(pos)
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for v in interior:
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nbrs = list(G_sage.neighbor_iterator(v))
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sx = sum(pos[w][0] for w in nbrs) / len(nbrs)
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sy = sum(pos[w][1] for w in nbrs) / len(nbrs)
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new_pos[v] = (sx, sy)
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pos = new_pos
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return pos
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OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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C = ['#dc2626', '#16a34a', '#2563eb']
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GRAY = '#9ca3af'
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DARK = '#374151'
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HIGHLIGHT = '#fef3c7'
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SHADE = '#fef3c7'
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def dual_of(G):
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faces = G.faces()
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edge_to_faces = {}
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for fi, face in enumerate(faces):
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for u, v in face:
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e = frozenset((u, v))
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edge_to_faces.setdefault(e, []).append(fi)
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dual_edges = []
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for e, fs in edge_to_faces.items():
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if len(fs) == 2:
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dual_edges.append((fs[0], fs[1]))
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return Graph(dual_edges, multiedges=False, loops=False)
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def apply_reduction(G, face, i, v_n_label):
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boundary = [u for (u, v) in face]
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if len(set(boundary)) != 5:
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return None
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A = []
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for B_k in boundary:
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outer = [w for w in G.neighbor_iterator(B_k) if w not in boundary]
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if len(outer) != 1:
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return None
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A.append(outer[0])
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if len(set(A)) != 5:
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return None
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if A[(i + 3) % 5] == A[(i + 4) % 5]:
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return None
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H = G.copy()
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for v in boundary:
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H.delete_vertex(v)
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H.add_vertex(v_n_label)
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side_0 = (v_n_label, A[i % 5])
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spike = (v_n_label, A[(i + 1) % 5])
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side_1 = (v_n_label, A[(i + 2) % 5])
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merged = (A[(i + 3) % 5], A[(i + 4) % 5])
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H.add_edges([side_0, spike, side_1, merged])
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if H.has_multiple_edges() or H.has_loops():
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return None
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if not H.is_planar(set_embedding=True):
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return None
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if not all(H.degree(v) == 3 for v in H.vertex_iterator()):
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return None
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named = {
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'spike': frozenset(spike),
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'side_0': frozenset(side_0),
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'side_1': frozenset(side_1),
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'merged': frozenset(merged),
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}
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return H, named, boundary, A
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def proper_3_edge_colorings(G):
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edges = list(G.edges(labels=False))
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n_edges = len(edges)
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adj = [[] for _ in range(n_edges)]
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for i in range(n_edges):
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u, v = edges[i][0], edges[i][1]
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for j in range(i):
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x, y = edges[j][0], edges[j][1]
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if u in (x, y) or v in (x, y):
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adj[i].append(j)
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adj[j].append(i)
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coloring = [-1] * n_edges
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def back(k):
|
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if k == n_edges:
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yield tuple(coloring)
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return
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for c in range(3):
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if all(coloring[j] != c for j in adj[k]):
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coloring[k] = c
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yield from back(k + 1)
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coloring[k] = -1
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return edges, back(0)
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def kempe_cycle(edges, coloring, start_idx, color_pair):
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a, b = color_pair
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in_sub = [i for i in range(len(edges)) if coloring[i] in (a, b)]
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if start_idx not in in_sub:
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return None
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visited = {start_idx}
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stack = [start_idx]
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while stack:
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cur = stack.pop()
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u, v = edges[cur][0], edges[cur][1]
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for j in in_sub:
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||||
if j in visited:
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continue
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x, y = edges[j][0], edges[j][1]
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if u in (x, y) or v in (x, y):
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visited.add(j)
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stack.append(j)
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return visited
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|
||||
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def matches_property(edges, col, named):
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idx = {}
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for ii, e in enumerate(edges):
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es = frozenset((e[0], e[1]))
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for role, ns in named.items():
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if es == ns:
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idx[role] = ii
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if len(idx) != 4:
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return False
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c_spike = col[idx['spike']]
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c_merged = col[idx['merged']]
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if c_spike != c_merged:
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return False
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c_s0 = col[idx['side_0']]
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c_s1 = col[idx['side_1']]
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kc0 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s0))
|
||||
if idx['side_0'] not in kc0 or idx['merged'] not in kc0:
|
||||
return False
|
||||
kc1 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s1))
|
||||
if idx['side_1'] not in kc1 or idx['merged'] not in kc1:
|
||||
return False
|
||||
return True
|
||||
|
||||
|
||||
def find_first_match():
|
||||
"""Iterate over (G, face, i_red, coloring) and return the first hit."""
|
||||
for G in graphs.triangulations(14, minimum_degree=5):
|
||||
if not G.is_planar(set_embedding=True):
|
||||
continue
|
||||
D = dual_of(G)
|
||||
D.is_planar(set_embedding=True)
|
||||
for face in D.faces():
|
||||
if len(face) != 5:
|
||||
continue
|
||||
for i_red in range(5):
|
||||
res = apply_reduction(D, face, i_red, '__v_n_1__')
|
||||
if res is None:
|
||||
continue
|
||||
H, named, boundary, A = res
|
||||
edges, gen = proper_3_edge_colorings(H)
|
||||
for col in gen:
|
||||
if matches_property(edges, col, named):
|
||||
coloring_dict = {frozenset((e[0], e[1])): c
|
||||
for e, c in zip(edges, col)}
|
||||
return G, D, face, i_red, H, named, boundary, A, coloring_dict
|
||||
return None
|
||||
|
||||
|
||||
def draw_graph(ax, G, pos, *, coloring=None, protected=None,
|
||||
shade_vertices=None, vn_labels=None):
|
||||
if shade_vertices:
|
||||
poly = [pos[v] for v in shade_vertices]
|
||||
ax.add_patch(Polygon(poly, closed=True, facecolor=SHADE,
|
||||
edgecolor='none', zorder=0))
|
||||
protected = protected or set()
|
||||
vn_labels = vn_labels or {}
|
||||
for u, v, _ in G.edges():
|
||||
e = frozenset([u, v])
|
||||
c = C[coloring[e]] if (coloring is not None and e in coloring) else GRAY
|
||||
lw = 3.8 if e in protected else 1.4
|
||||
(x0, y0), (x1, y1) = pos[u], pos[v]
|
||||
ax.plot([x0, x1], [y0, y1], color=c, lw=lw, zorder=2)
|
||||
for v in G.vertices(sort=False):
|
||||
x, y = pos[v]
|
||||
if v in vn_labels:
|
||||
ax.scatter(x, y, s=320, color=HIGHLIGHT, marker='s',
|
||||
edgecolors='black', linewidths=1.2, zorder=4)
|
||||
ax.annotate(vn_labels[v], (x, y),
|
||||
textcoords='offset points', xytext=(16, 16),
|
||||
ha='left', fontsize=14, fontweight='bold',
|
||||
color=DARK, zorder=6,
|
||||
bbox=dict(boxstyle='round,pad=0.2', fc='white',
|
||||
ec=DARK, lw=0.6))
|
||||
else:
|
||||
ax.scatter(x, y, s=60, color=DARK, zorder=3)
|
||||
ax.set_aspect('equal')
|
||||
ax.axis('off')
|
||||
|
||||
|
||||
def main():
|
||||
print("Searching for the first match at n = 14 ...")
|
||||
result = find_first_match()
|
||||
if result is None:
|
||||
print("No match found at n = 14.")
|
||||
return
|
||||
G14, D, face, i_red, H1, named1, boundary1, A1, coloring1 = result
|
||||
print(f"Found at i_red = {i_red}")
|
||||
print(f" G (n=14): |V|={G14.order()}, |E|={G14.size()}, "
|
||||
f"min_deg={min(G14.degree())}")
|
||||
print(f" D = G': |V|={D.order()}, |E|={D.size()}")
|
||||
print(f" H_1: |V|={H1.order()}, |E|={H1.size()}")
|
||||
|
||||
# Relabel H_1 in place so all vertex labels are comparable integers
|
||||
# (Sage's planar layout and face enumeration need comparable labels).
|
||||
# Translate coloring1 and named1 accordingly.
|
||||
H1_relabel_map = {v: i for i, v in enumerate(H1.vertex_iterator())}
|
||||
H1.relabel(perm=H1_relabel_map, inplace=True)
|
||||
vn1_int = H1_relabel_map['__v_n_1__']
|
||||
coloring1 = {frozenset(H1_relabel_map[u] for u in e): c
|
||||
for e, c in coloring1.items()}
|
||||
named1 = {role: frozenset(H1_relabel_map[u] for u in e)
|
||||
for role, e in named1.items()}
|
||||
|
||||
D.is_planar(set_embedding=True)
|
||||
D_layout = tutte_layout(D, avoid_verts=set(u for (u, v) in face))
|
||||
H1.is_planar(set_embedding=True)
|
||||
H1_layout = tutte_layout(H1, avoid_verts={vn1_int})
|
||||
|
||||
boundary_face_verts = [u for (u, v) in face]
|
||||
fig, ax = plt.subplots(figsize=(8, 8))
|
||||
draw_graph(ax, D, D_layout, shade_vertices=boundary_face_verts)
|
||||
fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step0.png'),
|
||||
dpi=170, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print("Wrote fig_alg_step0.png")
|
||||
|
||||
E = set(named1.values())
|
||||
fig, ax = plt.subplots(figsize=(8, 8))
|
||||
draw_graph(ax, H1, H1_layout, coloring=coloring1, protected=E,
|
||||
vn_labels={vn1_int: '$v_n^{(1)}$'})
|
||||
fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step1.png'),
|
||||
dpi=170, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print("Wrote fig_alg_step1.png")
|
||||
|
||||
# ----- Step 2: try to continue -----
|
||||
H1.is_planar(set_embedding=True)
|
||||
chosen2 = None
|
||||
for face2 in H1.faces():
|
||||
if len(face2) != 5:
|
||||
continue
|
||||
boundary2 = [u for (u, v) in face2]
|
||||
boundary2_edges = [frozenset([u, v]) for (u, v) in face2]
|
||||
externals2 = []
|
||||
A2 = []
|
||||
valid_face = True
|
||||
for B_k in boundary2:
|
||||
outer = [w for w in H1.neighbor_iterator(B_k) if w not in boundary2]
|
||||
if len(outer) != 1:
|
||||
valid_face = False
|
||||
break
|
||||
externals2.append(frozenset([B_k, outer[0]]))
|
||||
A2.append(outer[0])
|
||||
if not valid_face:
|
||||
continue
|
||||
if any(e in E for e in boundary2_edges + externals2):
|
||||
continue
|
||||
# find valid i_t
|
||||
f_vec = [coloring1[e] for e in externals2]
|
||||
for i_t in range(5):
|
||||
if f_vec[(i_t + 3) % 5] != f_vec[(i_t + 4) % 5]:
|
||||
continue
|
||||
if len({f_vec[i_t], f_vec[(i_t + 1) % 5], f_vec[(i_t + 2) % 5]}) != 3:
|
||||
continue
|
||||
if A2[(i_t + 3) % 5] == A2[(i_t + 4) % 5]:
|
||||
continue
|
||||
chosen2 = (face2, i_t, boundary2, externals2, A2)
|
||||
break
|
||||
if chosen2 is not None:
|
||||
break
|
||||
|
||||
if chosen2 is None:
|
||||
# algorithm terminates at H_1
|
||||
fig, ax = plt.subplots(figsize=(8, 8))
|
||||
ax.text(0.5, 0.5,
|
||||
"Algorithm terminates at $H_1$:\n"
|
||||
"no pentagonal face of $H_1$ has all\n"
|
||||
"ten incident edges outside $E$.",
|
||||
ha='center', va='center', fontsize=18, color=DARK,
|
||||
transform=ax.transAxes,
|
||||
bbox=dict(boxstyle='round,pad=0.6', fc=HIGHLIGHT,
|
||||
ec=DARK, lw=1.0))
|
||||
ax.set_aspect('equal')
|
||||
ax.axis('off')
|
||||
fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
|
||||
dpi=170, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print("Wrote fig_alg_step2.png (termination card)")
|
||||
print(" Algorithm terminates at H_1: no safe pentagonal face.")
|
||||
return
|
||||
|
||||
face2, i_t, boundary2, externals2, A2 = chosen2
|
||||
print(f"Step 2: safe face found, i_t = {i_t}")
|
||||
|
||||
H2 = H1.copy()
|
||||
for v in boundary2:
|
||||
H2.delete_vertex(v)
|
||||
# use a fresh int label for v_n^(2)
|
||||
v_n_2 = max(H1.vertices(sort=False)) + 1
|
||||
H2.add_vertex(v_n_2)
|
||||
side_0_2 = (v_n_2, A2[i_t])
|
||||
spike_2 = (v_n_2, A2[(i_t + 1) % 5])
|
||||
side_1_2 = (v_n_2, A2[(i_t + 2) % 5])
|
||||
merged_2 = (A2[(i_t + 3) % 5], A2[(i_t + 4) % 5])
|
||||
H2.add_edges([side_0_2, spike_2, side_1_2, merged_2])
|
||||
H2.is_planar(set_embedding=True)
|
||||
|
||||
coloring2 = {e: c for e, c in coloring1.items()
|
||||
if not any(u in boundary2 for u in e)}
|
||||
coloring2[frozenset(side_0_2)] = coloring1[externals2[i_t]]
|
||||
coloring2[frozenset(spike_2)] = coloring1[externals2[(i_t + 1) % 5]]
|
||||
coloring2[frozenset(side_1_2)] = coloring1[externals2[(i_t + 2) % 5]]
|
||||
coloring2[frozenset(merged_2)] = coloring1[externals2[(i_t + 3) % 5]]
|
||||
|
||||
E |= {frozenset(side_0_2), frozenset(spike_2),
|
||||
frozenset(side_1_2), frozenset(merged_2)}
|
||||
|
||||
H2_layout = tutte_layout(H2, avoid_verts={vn1_int, v_n_2})
|
||||
fig, ax = plt.subplots(figsize=(8, 8))
|
||||
draw_graph(ax, H2, H2_layout, coloring=coloring2, protected=E,
|
||||
vn_labels={vn1_int: '$v_n^{(1)}$',
|
||||
v_n_2: '$v_n^{(2)}$'})
|
||||
fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
|
||||
dpi=170, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f"Wrote fig_alg_step2.png: H_2 with |V|={H2.order()}, "
|
||||
f"|E|={H2.size()}, |protected|={len(E)}")
|
||||
|
||||
# --- continue running to completion, checking Kempe condition each step --
|
||||
print()
|
||||
print("=" * 72)
|
||||
print("Running algorithm to completion, checking chord-apex + Kempe at "
|
||||
"each step.")
|
||||
print("=" * 72)
|
||||
# Step 1 status (by construction this is the matching coloring)
|
||||
cond1 = check_step_conditions(H1, coloring1, named1)
|
||||
print(f" step t = 1: |V|={H1.order():>3}, |E_graph|={H1.size():>3}, "
|
||||
f"|E_prot|= 4 (initial)"
|
||||
f" | chord-apex: {cond1['chord_apex']}, "
|
||||
f"side_0-Kempe: {cond1['kc_side_0']}, "
|
||||
f"side_1-Kempe: {cond1['kc_side_1']}")
|
||||
run_to_completion_from(H2, coloring2, E,
|
||||
{'spike': frozenset(spike_2),
|
||||
'side_0': frozenset(side_0_2),
|
||||
'side_1': frozenset(side_1_2),
|
||||
'merged': frozenset(merged_2)},
|
||||
start_t=2)
|
||||
|
||||
|
||||
def check_step_conditions(H, coloring, named):
|
||||
"""Given an H_t and the *just-added* spike/side_0/side_1/merged, check
|
||||
whether chord-apex and the two Kempe-cycle conditions hold."""
|
||||
edges = list(H.edges(labels=False))
|
||||
edges_fs = [frozenset((u, v)) for (u, v) in edges]
|
||||
col = [coloring[e] for e in edges_fs]
|
||||
idx = {role: edges_fs.index(e) for role, e in named.items()}
|
||||
c_spike = col[idx['spike']]
|
||||
c_merged = col[idx['merged']]
|
||||
chord_apex = (c_spike == c_merged)
|
||||
if not chord_apex:
|
||||
return {'chord_apex': False, 'kc_side_0': False, 'kc_side_1': False}
|
||||
c_s0 = col[idx['side_0']]
|
||||
c_s1 = col[idx['side_1']]
|
||||
kc0 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s0))
|
||||
kc1 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s1))
|
||||
kc_side_0 = (idx['side_0'] in kc0 and idx['merged'] in kc0)
|
||||
kc_side_1 = (idx['side_1'] in kc1 and idx['merged'] in kc1)
|
||||
return {'chord_apex': True, 'kc_side_0': kc_side_0, 'kc_side_1': kc_side_1}
|
||||
|
||||
|
||||
def find_safe_face(H, protected):
|
||||
"""Return (face, externals, A) for some safe pentagonal face avoiding
|
||||
`protected`, or None."""
|
||||
for face in H.faces():
|
||||
if len(face) != 5:
|
||||
continue
|
||||
boundary = [u for (u, v) in face]
|
||||
boundary_edges = [frozenset([u, v]) for (u, v) in face]
|
||||
externals = []
|
||||
A = []
|
||||
valid = True
|
||||
for B_k in boundary:
|
||||
outer = [w for w in H.neighbor_iterator(B_k) if w not in boundary]
|
||||
if len(outer) != 1:
|
||||
valid = False
|
||||
break
|
||||
externals.append(frozenset([B_k, outer[0]]))
|
||||
A.append(outer[0])
|
||||
if not valid:
|
||||
continue
|
||||
if any(e in protected for e in boundary_edges + externals):
|
||||
continue
|
||||
return face, boundary, externals, A
|
||||
return None
|
||||
|
||||
|
||||
def run_to_completion_from(H, coloring, E, last_named, start_t):
|
||||
"""Continue iterating from H_{start_t}. The 'last_named' dict carries
|
||||
the spike/side/merged of step `start_t` so we can report its Kempe
|
||||
status. Print a row per step."""
|
||||
t = start_t
|
||||
print(f" step t = {t}: |V|={H.order():>3}, |E_graph|={H.size():>3}, "
|
||||
f"|E_prot|={len(E):>3}", end='')
|
||||
cond = check_step_conditions(H, coloring, last_named)
|
||||
print(f" | chord-apex: {cond['chord_apex']}, "
|
||||
f"side_0-Kempe: {cond['kc_side_0']}, "
|
||||
f"side_1-Kempe: {cond['kc_side_1']}")
|
||||
|
||||
while True:
|
||||
H.is_planar(set_embedding=True)
|
||||
res = find_safe_face(H, E)
|
||||
if res is None:
|
||||
print(f" step t = {t + 1}: no safe pentagonal face --> "
|
||||
f"algorithm terminates at H_{t}.")
|
||||
return
|
||||
face, boundary, externals, A = res
|
||||
f_vec = [coloring[e] for e in externals]
|
||||
i_t = None
|
||||
for i in range(5):
|
||||
if f_vec[(i + 3) % 5] != f_vec[(i + 4) % 5]:
|
||||
continue
|
||||
if len({f_vec[i], f_vec[(i + 1) % 5], f_vec[(i + 2) % 5]}) != 3:
|
||||
continue
|
||||
if A[(i + 3) % 5] == A[(i + 4) % 5]:
|
||||
continue
|
||||
i_t = i
|
||||
break
|
||||
if i_t is None:
|
||||
print(f" step t = {t + 1}: f = {f_vec}, no valid index --> "
|
||||
f"terminate (Lemma 2.4 violation? Probably a parallel-edge "
|
||||
f"or other degenerate case).")
|
||||
return
|
||||
t += 1
|
||||
v_n_new = max(H.vertices(sort=False)) + 1 if all(
|
||||
isinstance(v, int) for v in H.vertex_iterator()) else f'vn{t}'
|
||||
H_new = H.copy()
|
||||
for v in boundary:
|
||||
H_new.delete_vertex(v)
|
||||
H_new.add_vertex(v_n_new)
|
||||
side_0 = (v_n_new, A[i_t])
|
||||
spike = (v_n_new, A[(i_t + 1) % 5])
|
||||
side_1 = (v_n_new, A[(i_t + 2) % 5])
|
||||
merged = (A[(i_t + 3) % 5], A[(i_t + 4) % 5])
|
||||
H_new.add_edges([side_0, spike, side_1, merged])
|
||||
H = H_new
|
||||
coloring = {e: c for e, c in coloring.items()
|
||||
if not any(u in boundary for u in e)}
|
||||
coloring[frozenset(side_0)] = coloring[externals[i_t]] \
|
||||
if frozenset(externals[i_t]) in coloring else f_vec[i_t]
|
||||
# safer: directly use f_vec
|
||||
coloring[frozenset(side_0)] = f_vec[i_t]
|
||||
coloring[frozenset(spike)] = f_vec[(i_t + 1) % 5]
|
||||
coloring[frozenset(side_1)] = f_vec[(i_t + 2) % 5]
|
||||
coloring[frozenset(merged)] = f_vec[(i_t + 3) % 5]
|
||||
named = {
|
||||
'spike': frozenset(spike),
|
||||
'side_0': frozenset(side_0),
|
||||
'side_1': frozenset(side_1),
|
||||
'merged': frozenset(merged),
|
||||
}
|
||||
E |= set(named.values())
|
||||
cond = check_step_conditions(H, coloring, named)
|
||||
print(f" step t = {t}: |V|={H.order():>3}, |E_graph|={H.size():>3}, "
|
||||
f"|E_prot|={len(E):>3}, i_t = {i_t}", end='')
|
||||
print(f" | chord-apex: {cond['chord_apex']}, "
|
||||
f"side_0-Kempe: {cond['kc_side_0']}, "
|
||||
f"side_1-Kempe: {cond['kc_side_1']}")
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
main()
|
||||
Reference in New Issue
Block a user