didericis
435f055d82
New paper introducing the dual (inner/weak dual) of a maximal planar graph, dual depth (BFS-derived min level over a face's vertices), and a Tait-based framing of a minimal 4CT counterexample via nested level duals. Includes a dual-depth figure and its generator. Shelved per closing note in favour of an alternative approach. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
155 lines
6.2 KiB
Python
155 lines
6.2 KiB
Python
"""Draw a diagram illustrating dual depth.
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We build a concentric ("stacked rings") triangulation G with a single level
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source S = {0} at the centre, so the vertices fall into clean BFS levels
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0, 1, 2, 3 by radius. We then overlay the inner (weak) dual G' -- one dual
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vertex per bounded triangular face -- and colour each dual vertex by its dual
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depth: the minimum level among the three vertices of the corresponding face.
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The construction makes all three attainable dual depths (0, 1, 2) appear; the
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outer face (the level-3 triangle) is excluded from the inner dual.
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"""
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import math
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import os
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import networkx as nx
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import matplotlib.pyplot as plt
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from matplotlib.lines import Line2D
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OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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# ---------------------------------------------------------------------------
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# Construct G: centre 0 (level 0) plus three triangular rings at radii 1,2,3.
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# Vertex j of every ring sits at angle 90 + 120*j degrees, so spokes are radial.
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# ---------------------------------------------------------------------------
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RINGS = 3 # ring 3 is the outer-face triangle
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pos = {0: (0.0, 0.0)}
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ring = {0: [0]} # ring index -> vertex ids
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nxt = 1
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for r in range(1, RINGS + 1):
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ids = []
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for j in range(3):
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ang = math.radians(90 + 120 * j)
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pos[nxt] = (r * math.cos(ang), r * math.sin(ang))
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ids.append(nxt)
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nxt += 1
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ring[r] = ids
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G = nx.Graph()
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G.add_nodes_from(pos)
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# centre fan
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for v in ring[1]:
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G.add_edge(0, v)
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# ring triangles
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for r in range(1, RINGS + 1):
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a, b, c = ring[r]
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G.add_edges_from([(a, b), (b, c), (c, a)])
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# radial spokes + annulus diagonals (inner_j -- outer_{j+1})
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for r in range(1, RINGS):
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inner, outer = ring[r], ring[r + 1]
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for j in range(3):
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G.add_edge(inner[j], outer[j]) # spoke
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G.add_edge(inner[j], outer[(j + 1) % 3]) # diagonal
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assert G.number_of_edges() == 3 * G.number_of_nodes() - 6, "not a triangulation"
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# ---------------------------------------------------------------------------
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# Levels: BFS distance from the source S = {0}.
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# ---------------------------------------------------------------------------
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S = {0}
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level = nx.multi_source_dijkstra_path_length(G, S)
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# ---------------------------------------------------------------------------
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# Bounded faces (the inner dual's vertices). Listed explicitly from the
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# construction; the outer face (ring 3 triangle) is omitted.
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# ---------------------------------------------------------------------------
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faces = []
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a, b, c = ring[1]
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faces += [(0, a, b), (0, b, c), (0, c, a)] # centre fan
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for r in range(1, RINGS):
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inn, out = ring[r], ring[r + 1]
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for j in range(3):
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i0, i1 = inn[j], inn[(j + 1) % 3]
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o1 = out[(j + 1) % 3]
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o0 = out[j]
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faces += [(i0, i1, o1), (i0, o1, o0)] # the two annulus triangles
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def dual_depth(face):
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return min(level[v] for v in face)
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# dual vertex positions = face centroids
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dpos = {}
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ddepth = {}
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for f in faces:
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cx = sum(pos[v][0] for v in f) / 3.0
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cy = sum(pos[v][1] for v in f) / 3.0
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dpos[f] = (cx, cy)
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ddepth[f] = dual_depth(f)
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# dual edges: two bounded faces sharing an edge of G
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edge_faces = {}
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for f in faces:
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for i in range(3):
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e = frozenset((f[i], f[(i + 1) % 3]))
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edge_faces.setdefault(e, []).append(f)
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dual_edges = [fs for fs in edge_faces.values() if len(fs) == 2]
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# ---------------------------------------------------------------------------
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# Draw.
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# ---------------------------------------------------------------------------
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LEVEL_COLOR = {0: '#1e293b', 1: '#475569', 2: '#94a3b8', 3: '#cbd5e1'}
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DEPTH_COLOR = {0: '#16a34a', 1: '#2563eb', 2: '#dc2626'}
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fig, ax = plt.subplots(figsize=(9, 9))
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# G edges (light) and vertices (labelled by level)
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nx.draw_networkx_edges(G, pos, ax=ax, edge_color='#d1d5db', width=1.4)
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for v, (x, y) in pos.items():
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ax.scatter([x], [y], s=520, color=LEVEL_COLOR[level[v]],
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edgecolors='black', linewidths=1.0, zorder=3)
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ax.text(x, y, f'{v}\n$\\ell{{=}}{level[v]}$', ha='center', va='center',
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color='white', fontsize=8.5, fontweight='bold', zorder=4)
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# dual edges (dashed) and dual vertices (squares, coloured by dual depth)
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for f0, f1 in dual_edges:
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(x0, y0), (x1, y1) = dpos[f0], dpos[f1]
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ax.plot([x0, x1], [y0, y1], color='#fb923c', lw=1.3, ls='--', zorder=2)
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for f, (x, y) in dpos.items():
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ax.scatter([x], [y], s=300, marker='s', color=DEPTH_COLOR[ddepth[f]],
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edgecolors='black', linewidths=0.8, zorder=5)
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ax.text(x, y, str(ddepth[f]), ha='center', va='center',
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color='white', fontsize=9, fontweight='bold', zorder=6)
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legend = [
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Line2D([0], [0], marker='o', color='w', label='$G$ vertex (label = level $\\ell$)',
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markerfacecolor='#475569', markeredgecolor='black', markersize=12),
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Line2D([0], [0], color='#fb923c', ls='--', lw=1.3, label="dual edge of $G'$"),
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Line2D([0], [0], marker='s', color='w', label='dual depth $\\delta = 0$',
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markerfacecolor=DEPTH_COLOR[0], markeredgecolor='black', markersize=11),
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Line2D([0], [0], marker='s', color='w', label='dual depth $\\delta = 1$',
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markerfacecolor=DEPTH_COLOR[1], markeredgecolor='black', markersize=11),
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Line2D([0], [0], marker='s', color='w', label='dual depth $\\delta = 2$',
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markerfacecolor=DEPTH_COLOR[2], markeredgecolor='black', markersize=11),
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]
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ax.legend(handles=legend, loc='upper left', fontsize=10, framealpha=0.95)
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ax.set_aspect('equal')
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ax.axis('off')
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ax.set_title("Dual depth in a stacked-ring triangulation $G$ with source $S=\\{0\\}$.\n"
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"Each bounded face carries a dual vertex (square) coloured by its dual "
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"depth\n$\\delta(d_f)=\\min_{v\\in V(f)}\\ell(v)$. "
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"The outer face (level-3 triangle) has no dual vertex.",
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fontsize=11)
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_dual_depth.png')
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fig.savefig(out, dpi=180, bbox_inches='tight')
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plt.close(fig)
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# console summary
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from collections import Counter
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print(f'n={G.number_of_nodes()} edges={G.number_of_edges()} '
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f'bounded faces={len(faces)} dual edges={len(dual_edges)}')
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print('dual depth distribution:', dict(sorted(Counter(ddepth.values()).items())))
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print(f'wrote {out}')
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