From 717c8cf5ea5c90c0129a65b84a103aaba1383e9f Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 16:26:21 -0400 Subject: [PATCH] coloring_nested_tire_graphs: relax Def 1.5 to allow non-simple inner-boundary walks; drop (R2) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The previous reading of Def 1.5 had B_in be a simple cycle on V(O). This forced an artificial (R2) hypothesis on Lemma 1.7 ruling out multi-hole topology of R_{C'}. But the multi-hole structure is already captured naturally by the inner outerplanar graph O itself: when O is not 2-connected (has a bridge, cut-vertex, or multiple components), its outer-face boundary is a closed walk rather than a simple cycle, and that walk traverses bridges twice and visits cut- vertices multiple times. Changes: - Definition 1.5: B_in is now defined as the closed walk in O that traces O's outer-face boundary in the inherited embedding. It is a simple cycle when O is 2-connected and a non-simple closed walk in general. B_out remains a simple cycle (or degenerate single vertex). - Lemma 1.7: (R2) hypothesis removed. Only (R1) (manifold) remains. The proof of the boundary structure is rewritten: we identify B_out as the source-side boundary cycle, O as G[V_{C'} ∩ L_{d+1}] (outerplanar by Lemma 2.6 of [bauerfeld-pds]), and B_in as O's outer-face boundary walk. Multi-hole topology of R_{C'} is now captured by O being non-2-connected or disconnected, without needing an extra constraint. - Remark 1.10 (was R1+R2): now Remark 1.10 (R1-when), explaining the pinch obstruction for (R1) and explicitly noting that multi-hole topology does NOT require an additional hypothesis under the new Definition 1.5. Empirical motivation: in the brute-force search over maximal planar graphs at n in [7, 11] (30,587 components from all single-vertex sources), 171 components at depth 1 had R_{C'} with 3 boundary cycles in the surface-classification sense (n_bdry > 2). Each is a valid tire graph under the relaxed definition: B_out is the level-d cycle, O is the level-(d+1) outerplanar subgraph (non-2-connected when this occurs), and B_in is its outer-face boundary walk. Co-Authored-By: Claude Opus 4.7 --- .../experiments/check_r2.py | 253 ++++++++++++++++++ .../experiments/draw_r2_violation_example.py | 191 +++++++++++++ .../experiments/draw_r2_violation_v2.py | 181 +++++++++++++ .../experiments/r2_violation_real_example.png | Bin 0 -> 99666 bytes .../experiments/r2_violation_schematic.png | Bin 0 -> 103971 bytes .../experiments/r2_violation_schematic.py | 154 +++++++++++ .../experiments/r2_violation_v2.png | Bin 0 -> 100596 bytes papers/coloring_nested_tire_graphs/paper.aux | 9 +- papers/coloring_nested_tire_graphs/paper.log | 53 ++-- papers/coloring_nested_tire_graphs/paper.pdf | Bin 486337 -> 486084 bytes papers/coloring_nested_tire_graphs/paper.tex | 183 +++++++------ 11 files changed, 903 insertions(+), 121 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/check_r2.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_example.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_v2.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/r2_violation_real_example.png create mode 100644 papers/coloring_nested_tire_graphs/experiments/r2_violation_schematic.png create mode 100644 papers/coloring_nested_tire_graphs/experiments/r2_violation_schematic.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/r2_violation_v2.png diff --git a/papers/coloring_nested_tire_graphs/experiments/check_r2.py b/papers/coloring_nested_tire_graphs/experiments/check_r2.py new file mode 100644 index 0000000..07c2278 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/check_r2.py @@ -0,0 +1,253 @@ +"""Brute-force search for (R1)/(R2) violations in small maximal planar +graphs. + +For each maximal planar graph G on n vertices (with delta(G) >= 3), +for each single-vertex source v_0 in V(G), compute: + - BFS depths from v_0. + - For each face, its dual depth (= min BFS level of vertices). + - For each depth d, connected components of the dual subgraph G'_d. + - For each component, check (R1) manifold property and count (R2) + boundary components. + +Report: + - Per depth d, count of components and count of (R1)-violators, + (R2)-violators, both. + +Run: sage -python experiments/check_r2.py [n_min] [n_max] +""" +import sys +import json +from collections import defaultdict + +from sage.all import Graph +from sage.graphs.graph_generators import graphs + + +def face_verts(face): + """face = list of edges [(u, v), (v, w), ...]. Return ordered vertex + cycle [u, v, w, ...].""" + if not face: + return [] + verts = [face[0][0]] + for e in face: + verts.append(e[1]) + return verts[:-1] # last = first + + +def face_edges_set(face): + return {frozenset(e) for e in face} + + +def find_components(face_depths, target_d): + """face_depths: list of (face, depth, vertex_set). + Returns list of components; each component = list of (face_idx, face_depth_entry).""" + cands = [(i, fd) for i, fd in enumerate(face_depths) if fd[1] == target_d] + n = len(cands) + if n == 0: + return [] + edge_to_idxs = defaultdict(list) + for k, (i, fd) in enumerate(cands): + for e in face_edges_set(fd[0]): + edge_to_idxs[e].append(k) + adj = [[] for _ in range(n)] + for idxs in edge_to_idxs.values(): + if len(idxs) >= 2: + for a in idxs: + for b in idxs: + if a != b: + adj[a].append(b) + visited = [False] * n + comps = [] + for s in range(n): + if visited[s]: + continue + stack = [s] + comp = [] + while stack: + x = stack.pop() + if visited[x]: + continue + visited[x] = True + comp.append(cands[x]) + for y in adj[x]: + if not visited[y]: + stack.append(y) + comps.append(comp) + return comps + + +def check_component(comp, G, emb): + """For a connected component of same-depth faces, check R1 (manifold) + and count boundary components (R2).""" + # boundary edges: edges of G appearing in exactly one face of comp + edge_count = defaultdict(int) + for (_, fd) in comp: + for e in face_edges_set(fd[0]): + edge_count[e] += 1 + bdry_edges = {e for e, c in edge_count.items() if c == 1} + if not bdry_edges: + # No boundary (closed surface). In a planar graph this shouldn't happen. + return {'is_manifold': False, 'n_boundary': 0, 'reason': 'no boundary'} + + # R1 check: at each vertex v in V(comp), count how many faces of comp + # are incident to v, and check whether they form a contiguous arc in + # v's rotation in emb. + comp_face_edges = [] + for (_, fd) in comp: + comp_face_edges.append(face_edges_set(fd[0])) + vertices = set() + for fes in comp_face_edges: + for e in fes: + vertices.update(e) + + is_manifold = True + pinch_vertices = [] + for v in vertices: + rot = emb[v] # cyclic neighbour order + # The faces incident to v in G are the triangles {v, rot[i], rot[i+1]}. + # Check which ones are in comp. + in_comp = [] + for i in range(len(rot)): + u = rot[i] + w = rot[(i + 1) % len(rot)] + face_e = {frozenset({v, u}), frozenset({u, w}), frozenset({v, w})} + if any(face_e == fes for fes in comp_face_edges): + in_comp.append(True) + else: + in_comp.append(False) + # Count "runs" of True in cyclic sequence + n_true = sum(in_comp) + if n_true == 0: + continue + if n_true == len(in_comp): + continue # entire rotation in comp, vertex is interior + # Count transitions from False to True (cyclic) + n_runs = 0 + for i in range(len(in_comp)): + if in_comp[i] and not in_comp[(i - 1) % len(in_comp)]: + n_runs += 1 + if n_runs > 1: + is_manifold = False + pinch_vertices.append(v) + + # R2 check: count boundary components. + # Build bdry_graph: undirected graph on vertices touching bdry_edges, + # edges = bdry_edges. + bdry_graph = defaultdict(set) + for e in bdry_edges: + u, w = list(e) + bdry_graph[u].add(w) + bdry_graph[w].add(u) + # Count connected components + seen = set() + n_components = 0 + for v in bdry_graph: + if v in seen: + continue + n_components += 1 + stack = [v] + while stack: + x = stack.pop() + if x in seen: + continue + seen.add(x) + for y in bdry_graph[x]: + if y not in seen: + stack.append(y) + return {'is_manifold': is_manifold, + 'n_boundary': n_components, + 'pinch_vertices': pinch_vertices, + 'n_bdry_edges': len(bdry_edges), + 'n_faces': len(comp)} + + +def search(n_min, n_max, max_violations=5, verbose=False): + r1_violations = [] + r2_violations = [] + summary = defaultdict(int) + + for n in range(n_min, n_max + 1): + ntri = 0 + for G in graphs.triangulations(n): + ntri += 1 + G.is_planar(set_embedding=True) + emb = G.get_embedding() + faces = G.faces(embedding=emb) + for v_source in G.vertices(): + dist = G.shortest_path_lengths(v_source) + face_depths = [] + for face in faces: + verts = set() + for u, w in face: + verts.add(u); verts.add(w) + d = min(dist[u] for u in verts) + face_depths.append((face, d, verts)) + max_d = max(fd[1] for fd in face_depths) + for target_d in range(max_d + 1): + comps = find_components(face_depths, target_d) + for comp_idx, comp in enumerate(comps): + result = check_component(comp, G, emb) + summary[('component', target_d)] += 1 + if not result['is_manifold']: + summary[('r1_violation', target_d)] += 1 + if len(r1_violations) < max_violations: + r1_violations.append({ + 'n': n, 'source': int(v_source), + 'depth': target_d, + 'graph_edges': sorted([sorted(e) for e in G.edges(labels=False)]), + 'pinch_vertices': result['pinch_vertices'], + }) + if result['is_manifold'] and result['n_boundary'] > 2: + summary[('r2_violation_manifold', target_d)] += 1 + if len(r2_violations) < max_violations: + r2_violations.append({ + 'n': n, 'source': int(v_source), + 'depth': target_d, + 'n_boundary': result['n_boundary'], + 'graph_edges': sorted([sorted(e) for e in G.edges(labels=False)]), + }) + if verbose: + print(f"n={n}: scanned {ntri} triangulations") + sys.stdout.flush() + + return r1_violations, r2_violations, summary + + +def main(): + n_min = int(sys.argv[1]) if len(sys.argv) > 1 else 7 + n_max = int(sys.argv[2]) if len(sys.argv) > 2 else 12 + print(f"Searching maximal planar graphs n in [{n_min}, {n_max}]...") + r1s, r2s, summary = search(n_min, n_max, max_violations=5, verbose=True) + print() + print("=== Summary ===") + by_depth = defaultdict(lambda: {'components': 0, 'r1': 0, 'r2_manifold': 0}) + for (kind, d), c in summary.items(): + if kind == 'component': + by_depth[d]['components'] += c + elif kind == 'r1_violation': + by_depth[d]['r1'] += c + elif kind == 'r2_violation_manifold': + by_depth[d]['r2_manifold'] += c + print(f"{'depth':>6} {'components':>12} {'R1 viol':>10} {'R2 viol (manifold)':>20}") + for d in sorted(by_depth): + s = by_depth[d] + print(f"{d:>6} {s['components']:>12} {s['r1']:>10} {s['r2_manifold']:>20}") + + print() + if r1s: + print(f"=== R1 violations (first {len(r1s)}) ===") + for v in r1s: + print(f" n={v['n']}, source={v['source']}, depth={v['depth']}, " + f"pinches={v['pinch_vertices']}") + if r2s: + print(f"=== R2 violations (manifold, n_bdry > 2) (first {len(r2s)}) ===") + for v in r2s: + print(f" n={v['n']}, source={v['source']}, depth={v['depth']}, " + f"n_bdry={v['n_boundary']}") + print(f" edges: {v['graph_edges']}") + if not r1s and not r2s: + print("No R1 or R2 violations found in this range.") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_example.py b/papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_example.py new file mode 100644 index 0000000..139c7de --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_example.py @@ -0,0 +1,191 @@ +"""Draw an actual maximal planar graph exhibiting an (R2) violation. + +Uses the first such example found by experiments/check_r2.py: + n=10, source=4, depth=1, n_bdry=3. +""" +import os +import sys +from collections import defaultdict + +from sage.all import Graph +from sage.graphs.graph_generators import graphs + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +import matplotlib.pyplot as plt +import matplotlib.patches as patches + +# Edges from the first n=10 R2-violating triangulation found. +EDGES = [(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), + (2, 3), (2, 8), (2, 9), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8), + (3, 10), (4, 5), (5, 6), (6, 7), (6, 10), (7, 8), (7, 10), (8, 9)] +SOURCE = 4 +DEPTH = 1 + + +def main(): + G = Graph(EDGES) + G.is_planar(set_embedding=True) + emb = G.get_embedding() + dist = G.shortest_path_lengths(SOURCE) + print(f"BFS distances from {SOURCE}: {dict(dist)}") + + faces = G.faces(embedding=emb) + face_depths = [] + for face in faces: + verts = set() + for u, v in face: + verts.add(u); verts.add(v) + d = min(dist[v] for v in verts) + face_depths.append((face, d, verts)) + + # Find depth-DEPTH faces and their connected components + target_faces = [(i, fd) for i, fd in enumerate(face_depths) + if fd[1] == DEPTH] + print(f"Found {len(target_faces)} faces at depth {DEPTH}") + + edge_to_idxs = defaultdict(list) + for k, (i, fd) in enumerate(target_faces): + for u, v in fd[0]: + edge_to_idxs[frozenset({u, v})].append(k) + n = len(target_faces) + adj = [[] for _ in range(n)] + for idxs in edge_to_idxs.values(): + if len(idxs) >= 2: + for a in idxs: + for b in idxs: + if a != b: + adj[a].append(b) + visited = [False] * n + comps = [] + for s in range(n): + if visited[s]: + continue + stack = [s]; comp = [] + while stack: + x = stack.pop() + if visited[x]: continue + visited[x] = True; comp.append(target_faces[x]) + for y in adj[x]: + if not visited[y]: stack.append(y) + comps.append(comp) + print(f"Connected components at depth {DEPTH}: {len(comps)} sizes " + f"{[len(c) for c in comps]}") + # Pick the largest one + comp = max(comps, key=lambda c: len(c)) + + # Compute boundary edges of comp + edge_count = defaultdict(int) + comp_face_edges = [] + for _, fd in comp: + fe = {frozenset({u, v}) for u, v in fd[0]} + comp_face_edges.append(fe) + for e in fe: + edge_count[e] += 1 + bdry_edges = {e for e, c in edge_count.items() if c == 1} + + # Boundary connected components + bdry_graph = defaultdict(set) + for e in bdry_edges: + u, v = list(e) + bdry_graph[u].add(v) + bdry_graph[v].add(u) + seen = set() + bdry_cycles = [] + for v in bdry_graph: + if v in seen: continue + comp_v = set(); stack = [v] + while stack: + x = stack.pop() + if x in seen: continue + seen.add(x); comp_v.add(x) + for y in bdry_graph[x]: + if y not in seen: stack.append(y) + bdry_cycles.append(comp_v) + print(f"Boundary components: {len(bdry_cycles)}, vertex sets: {bdry_cycles}") + + # Compute layout using sage's planar layout + pos = G.layout(layout='planar') + + # Plot + fig, ax = plt.subplots(figsize=(9, 9)) + + # Fill in depth-1 component faces + for _, fd in comp: + face = fd[0] + cycle_verts = [edge[0] for edge in face] + poly = plt.Polygon([pos[v] for v in cycle_verts], + facecolor='#fff3cd', edgecolor='none', + alpha=0.55, zorder=0) + ax.add_patch(poly) + + # Draw all edges of G + for u, v in G.edges(labels=False): + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color='lightgray', linewidth=0.8, + zorder=1) + + # Highlight boundary edges of comp (in 3 different colors) + colors = ['#d62728', '#1f77b4', '#2ca02c'] + cyclic_label = ['boundary 1', 'boundary 2', 'boundary 3'] + for i, bc_verts in enumerate(bdry_cycles): + for e in bdry_edges: + if e <= bc_verts: + u, v = list(e) + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=colors[i % 3], + linewidth=3.0, zorder=2) + + # Draw vertices, color-coded by BFS level + level_colors = {0: 'black', 1: '#1f77b4', 2: '#d62728'} + for v in G.vertices(): + x, y = pos[v] + c = level_colors.get(dist[v], 'gray') + ax.plot(x, y, 'o', color=c, markersize=22, zorder=3) + ax.annotate(f"{v}", (x, y), color='white', ha='center', va='center', + fontsize=11, fontweight='bold', zorder=4) + # Add depth annotation + ax.annotate(f"$\\ell={dist[v]}$", (x, y), + xytext=(x + 0.05, y + 0.05), fontsize=9, zorder=4, + color='black', + bbox=dict(boxstyle='round,pad=0.15', + facecolor='white', edgecolor='none', + alpha=0.75)) + + # Legend + legend_items = [ + plt.Line2D([], [], marker='o', color='w', markerfacecolor='black', + markersize=12, label='source $v_0$ ($\\ell=0$)'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#1f77b4', + markersize=12, label='$\\ell=1$'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#d62728', + markersize=12, label='$\\ell=2$'), + plt.Line2D([], [], color='#fff3cd', linewidth=12, + label=f'$R_{{C^\\prime}}$ (depth-{DEPTH} component, ' + f'{len(comp)} faces)'), + plt.Line2D([], [], color='#d62728', linewidth=3, + label='boundary cycle 1'), + plt.Line2D([], [], color='#1f77b4', linewidth=3, + label='boundary cycle 2'), + plt.Line2D([], [], color='#2ca02c', linewidth=3, + label='boundary cycle 3'), + ] + ax.legend(handles=legend_items, loc='upper right', fontsize=9, + framealpha=0.95) + + ax.set_title(f"(R2) violation: depth-{DEPTH} component with " + f"3 boundary cycles\n(maximal planar graph, $n=10$, " + f"source $v_0={SOURCE}$)", + fontsize=12) + ax.set_aspect('equal') + ax.axis('off') + + out = os.path.join(HERE, 'r2_violation_real_example.png') + plt.savefig(out, dpi=160, bbox_inches='tight') + plt.close() + print(f"wrote {out}") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_v2.py b/papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_v2.py new file mode 100644 index 0000000..40c4e24 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_r2_violation_v2.py @@ -0,0 +1,181 @@ +"""Draw the n=10 R2-violating example with a planar layout that puts +source v_0=4 in the centre. We pick one of the depth-2 face triangles +({2,8,9} or {6,7,10}) as the outer face, which forces sage's planar +layout to put it on the outside and v_0 in the interior.""" +import os +import sys +from collections import defaultdict + +from sage.all import Graph + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +import matplotlib.pyplot as plt +import matplotlib.patches as patches +from matplotlib.path import Path + +EDGES = [(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), + (2, 3), (2, 8), (2, 9), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8), + (3, 10), (4, 5), (5, 6), (6, 7), (6, 10), (7, 8), (7, 10), (8, 9)] +SOURCE = 4 +DEPTH = 1 + + +def main(): + G = Graph(EDGES) + G.is_planar(set_embedding=True) + emb = G.get_embedding() + dist = G.shortest_path_lengths(SOURCE) + + # Use sage's planar layout with external_face = the depth-2 face {2, 8, 9}. + # The 'external_face' parameter in sage takes a face (list of edges). + # We supply edges (2,8), (8,9), (9,2) in cyclic order. + # Force external face to be the depth-2 triangle {2, 8, 9} by + # giving an edge of that triangle. + pos = G.layout(layout='planar', external_face=(2, 8)) + # Normalise positions + xs = [p[0] for p in pos.values()] + ys = [p[1] for p in pos.values()] + cx, cy = (max(xs) + min(xs)) / 2, (max(ys) + min(ys)) / 2 + scale = max(max(xs) - min(xs), max(ys) - min(ys)) + pos = {v: ((p[0] - cx) / scale, (p[1] - cy) / scale) + for v, p in pos.items()} + + # Compute depth-1 component + faces = G.faces(embedding=emb) + face_depths = [] + for face in faces: + verts = set() + for u, v in face: + verts.add(u); verts.add(v) + d = min(dist[v] for v in verts) + face_depths.append((face, d, verts)) + depth1_faces = [fd for fd in face_depths if fd[1] == DEPTH] + + # Boundary edges + edge_count = defaultdict(int) + for face, _, _ in depth1_faces: + for u, v in face: + edge_count[frozenset({u, v})] += 1 + bdry_edges = {e for e, c in edge_count.items() if c == 1} + + bdry_graph = defaultdict(set) + for e in bdry_edges: + u, v = list(e) + bdry_graph[u].add(v) + bdry_graph[v].add(u) + seen = set() + bdry_cycles = [] + for v in bdry_graph: + if v in seen: continue + cv = set(); stack = [v] + while stack: + x = stack.pop() + if x in seen: continue + seen.add(x); cv.add(x) + for y in bdry_graph[x]: + if y not in seen: stack.append(y) + bdry_cycles.append(cv) + # Sort: source-side first (= contains a level-1 vertex), then by min vertex + def cyc_key(c): + levels = {dist[v] for v in c} + return (min(levels), min(c)) + bdry_cycles.sort(key=cyc_key) + + print(f"Boundary cycles: {bdry_cycles}") + + # Plot + fig, ax = plt.subplots(figsize=(10, 9)) + + # Fill the depth-1 face union with one polygon: use the boundary curves + # Construct the outer boundary curve and the inner hole curves. + # The outer-facing boundary cycle (level d, source side) — but with + # external_face being a level-2 triangle, the "outer face" of the + # picture is the depth-2 triangle, and the depth-1 region is between. + # Easier: just fill each depth-1 triangle separately. + for face, _, _ in depth1_faces: + verts = [e[0] for e in face] + xy = [pos[v] for v in verts] + poly = plt.Polygon(xy, facecolor='#fff3cd', edgecolor='#e8d8a8', + linewidth=0.4, alpha=0.85, zorder=1) + ax.add_patch(poly) + + # Draw ALL non-boundary edges of G (interior and outside) in light grey + for u, v in G.edges(labels=False): + if frozenset({u, v}) in bdry_edges: + continue + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color='lightgray', linewidth=0.9, + zorder=2) + + # Boundary cycles in different colors + bdry_colors = ['#d62728', '#1f77b4', '#2ca02c'] + bdry_labels = [ + f'source-side boundary on $L_{{1}}$ (= $\\{{1, 3, 5\\}}$)', + f'inner boundary A on $L_{{2}}$', + f'inner boundary B on $L_{{2}}$', + ] + for i, cyc in enumerate(bdry_cycles): + for e in bdry_edges: + if e <= cyc: + u, v = list(e) + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=bdry_colors[i], + linewidth=3.5, zorder=3) + + # Draw vertices + level_colors = {0: 'black', 1: '#1f77b4', 2: '#d62728'} + for v in G.vertices(): + x, y = pos[v] + c = level_colors.get(dist[v], 'gray') + ax.plot(x, y, 'o', color=c, markersize=22, zorder=4) + ax.annotate(f"{v}", (x, y), color='white', ha='center', va='center', + fontsize=11, fontweight='bold', zorder=5) + + # Source label + sx, sy = pos[SOURCE] + ax.annotate(f'source $v_0={SOURCE}$\n($\\ell=0$)', + xy=(sx, sy), xytext=(sx, sy - 0.10), + fontsize=10, ha='center', va='top', color='black', zorder=6, + bbox=dict(boxstyle='round,pad=0.25', + facecolor='white', edgecolor='gray', alpha=0.95)) + + legend_items = [ + plt.Line2D([], [], marker='o', color='w', markerfacecolor='black', + markersize=11, label=f'source ($\\ell=0$)'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#1f77b4', + markersize=11, label='$\\ell=1$'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#d62728', + markersize=11, label='$\\ell=2$'), + patches.Patch(facecolor='#fff3cd', edgecolor='#e8d8a8', + label=f'$R_{{C^\\prime}}$ (depth-{DEPTH} component, ' + f'{len(depth1_faces)} triangles)'), + ] + for i, lbl in enumerate(bdry_labels): + legend_items.append(plt.Line2D([], [], color=bdry_colors[i], + linewidth=3.5, label=lbl)) + ax.legend(handles=legend_items, loc='upper right', fontsize=9, + framealpha=0.95) + + ax.set_title( + f'(R2) violation: depth-1 component with 3 boundary cycles\n' + f'planar embedding of n=10 maximal planar graph, ' + f'source $v_0={SOURCE}$ at centre', + fontsize=12) + ax.set_aspect('equal') + ax.axis('off') + pad = 0.08 + ax.set_xlim(min(p[0] for p in pos.values()) - pad, + max(p[0] for p in pos.values()) + pad) + ax.set_ylim(min(p[1] for p in pos.values()) - pad, + max(p[1] for p in pos.values()) + pad) + + out = os.path.join(HERE, 'r2_violation_v2.png') + plt.savefig(out, dpi=160, bbox_inches='tight') + plt.close() + print(f'wrote {out}') + + +if __name__ == '__main__': + 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zM%*;rdRNHKbEWg?y77Vu&{d(8*+%kSC28e4tax0Xt^)GE2&(J{mj)R`G;~Y0ZrPOv ztT~=oYyvHTO*T43APpqRQUNPUMIx%Ph6sC@a)^Zj{$s`=E?_7Q01-$HH6FWF9BD}p zn)C#|R?fPn1^^Q6dMW6ai_oh-aF0k+0AS32mkw02^S`S<5ZXlh-*t-$?fU)iBKoge z_upS{LdS{x-^(2;(fQvMjRK8f|GUnJp*h6=E;Jzw)G+@aFv}|UGs>H;WyJ;= RKi?PN5*LvXE)~@C`9DP9lAZtn literal 0 HcmV?d00001 diff --git a/papers/coloring_nested_tire_graphs/experiments/r2_violation_schematic.py b/papers/coloring_nested_tire_graphs/experiments/r2_violation_schematic.py new file mode 100644 index 0000000..9f6b755 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/r2_violation_schematic.py @@ -0,0 +1,154 @@ +"""Schematic illustration of an (R2)-violating tire component. + +A connected depth-d face component R_{C'} that has THREE boundary +components: one toward the source side (level d cycle), and two toward +the depth->d side (two distinct deeper lobes each enclosed by R_{C'}). + +This is the topological "pair of pants" / trinion: a planar 2-manifold +with three boundary curves, Euler characteristic -1. +""" +import math +import os +import matplotlib.pyplot as plt +import matplotlib.patches as patches +from matplotlib.patches import Circle, Polygon, FancyArrowPatch + + +def draw_r2_violation(filename): + fig, ax = plt.subplots(figsize=(8.5, 7)) + + # Outer boundary of R_{C'} -- a large ellipse-like blob + outer_r = 3.2 + + # Three holes + # Source hole on the left + source_center = (-1.6, 0.2) + source_r = 0.7 + + # Two deeper lobes on the right + lobe1_center = (1.3, 1.1) + lobe1_r = 0.55 + + lobe2_center = (1.3, -1.1) + lobe2_r = 0.55 + + # Draw the filled R_{C'} region: outer ellipse minus three discs. + # Using a Path with the outer circle and three reversed inner circles. + from matplotlib.path import Path + import numpy as np + + def circle_verts(cx, cy, r, n=80, reverse=False): + ts = np.linspace(0, 2 * np.pi, n, endpoint=False) + if reverse: + ts = ts[::-1] + return [(cx + r * np.cos(t), cy + r * np.sin(t)) for t in ts] + + outer_verts = circle_verts(0, 0, outer_r, n=120) + # Make outer ellipse instead of circle + outer_verts = [(1.4 * x, y) for (x, y) in outer_verts] + + inner1 = circle_verts(*source_center, source_r, n=60, reverse=True) + inner2 = circle_verts(*lobe1_center, lobe1_r, n=60, reverse=True) + inner3 = circle_verts(*lobe2_center, lobe2_r, n=60, reverse=True) + + verts = (outer_verts + [outer_verts[0]] + + inner1 + [inner1[0]] + + inner2 + [inner2[0]] + + inner3 + [inner3[0]]) + + codes = ([Path.MOVETO] + [Path.LINETO] * (len(outer_verts) - 1) + [Path.CLOSEPOLY] + + [Path.MOVETO] + [Path.LINETO] * (len(inner1) - 1) + [Path.CLOSEPOLY] + + [Path.MOVETO] + [Path.LINETO] * (len(inner2) - 1) + [Path.CLOSEPOLY] + + [Path.MOVETO] + [Path.LINETO] * (len(inner3) - 1) + [Path.CLOSEPOLY]) + + path = Path(verts, codes) + patch = patches.PathPatch(path, facecolor='#fff3cd', edgecolor='none', + alpha=0.85, zorder=1) + ax.add_patch(patch) + + # Draw the three boundary curves on top + def plot_curve(verts, color, lw=2.5, label_pt=None, label_text=None, + label_offset=(0, 0), label_color=None): + xs = [v[0] for v in verts] + [verts[0][0]] + ys = [v[1] for v in verts] + [verts[0][1]] + ax.plot(xs, ys, color=color, linewidth=lw, zorder=2) + if label_pt is not None and label_text is not None: + ax.annotate(label_text, xy=label_pt, + xytext=(label_pt[0] + label_offset[0], + label_pt[1] + label_offset[1]), + fontsize=11, ha='center', va='center', + color=label_color or color, zorder=3, + bbox=dict(boxstyle='round,pad=0.3', + edgecolor='none', facecolor='white', + alpha=0.85)) + + plot_curve(outer_verts, '#1f77b4', lw=2.8) + plot_curve(inner1, '#d62728', lw=2.5) + plot_curve(inner2, '#d62728', lw=2.5) + plot_curve(inner3, '#d62728', lw=2.5) + + # Source vertex marker in the source hole + sx, sy = source_center + ax.plot(sx, sy, 'o', color='black', markersize=12, zorder=4) + ax.annotate(r'$S = \{v_0\}$', xy=(sx, sy), + xytext=(sx, sy - 0.32), + fontsize=11, ha='center', va='top', zorder=4, + bbox=dict(boxstyle='round,pad=0.25', + edgecolor='gray', facecolor='white', alpha=0.95)) + + # Region labels + ax.annotate(r'$R_{C^\prime}$ (depth-$d$ region)', + xy=(-3.5, 2.6), fontsize=14, ha='left', va='center', + color='#9a7d2a', fontweight='bold') + + # Labels for the three boundary cycles + ax.annotate(r'outer boundary' + '\n' + r'on $L_{d}$ (source side)', + xy=(-1.6 + 0.7 + 0.05, 0.2), xytext=(-2.0, -1.6), + fontsize=10, ha='center', color='#d62728', + arrowprops=dict(arrowstyle='->', color='#d62728', lw=1.2)) + + ax.annotate(r'lobe boundary on $L_{d+1}$', + xy=(1.3 + 0.55 - 0.05, 1.1), xytext=(3.0, 2.4), + fontsize=10, ha='left', color='#d62728', + arrowprops=dict(arrowstyle='->', color='#d62728', lw=1.2)) + + ax.annotate(r'lobe boundary on $L_{d+1}$', + xy=(1.3 + 0.55 - 0.05, -1.1), xytext=(3.0, -2.4), + fontsize=10, ha='left', color='#d62728', + arrowprops=dict(arrowstyle='->', color='#d62728', lw=1.2)) + + ax.annotate(r'outer ellipse $=$' + '\n' + r'outer face of $\Pi_G$', + xy=(-3.5 * 1.4 + 0.2, 0), xytext=(-4.8, 0), + fontsize=10, ha='center', color='#1f77b4', + arrowprops=dict(arrowstyle='->', color='#1f77b4', lw=1.2)) + + # Lobe interior annotations + ax.annotate('depth-$> d$\nlobe A', + xy=lobe1_center, fontsize=10, ha='center', va='center', + color='#444', style='italic') + ax.annotate('depth-$> d$\nlobe B', + xy=lobe2_center, fontsize=10, ha='center', va='center', + color='#444', style='italic') + + # Title / caption text + ax.text(0, 3.4, + '(R2) violation: $R_{C^\\prime}$ has 3 boundary components', + fontsize=13, ha='center', fontweight='bold') + ax.text(0, -3.2, + r'$\chi(R_{C^\prime}) = V - E + F = 2 - n_{\mathrm{bdy}} = -1$', + fontsize=11, ha='center', color='#666') + + ax.set_xlim(-5.5, 5.5) + ax.set_ylim(-3.6, 3.7) + ax.set_aspect('equal') + ax.axis('off') + + plt.savefig(filename, dpi=160, bbox_inches='tight') + plt.close() + + +if __name__ == '__main__': + here = os.path.dirname(os.path.abspath(__file__)) + out = os.path.join(here, 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zN1oV~$NH8$VIhz1$rIO?r+c!{7RSr3I3<5N=;&D(+ZQH11f2H1zaxz9*~+%CNegUb zSDYl*7pJqO?nq<1()dtF;}?dqswa$EUzpCWxFd{TDU4EaAx-E>qt} zLps9PJA_dSGnt++etluO<$Fh*o(sz*f!O+Go$|LMOg{|ALK<5vC)X3^0a3V9 zy|#sk{UEFgVO-%feGh&>6z-Op9clFHj9Ivija_k|URYnAPHol^$KGEY(--Fffw)^C zb>!)}u_?qU5~l16b4wh?U-L}y=R7Xg_yS`fUXD(t)o1=ejpm=;X#O>h;VVN|o!>?pCNtjI%iA{(^((s;V`Q?9};ZaJdKd8v5vXlz>2#A7GLAj+!p3IOeNs~oV zBy*%d7RWp)!Cm*s0}npSlhVY33Op)UACBA{Vt>Q>`jI(HGSH<5nSSCzm-l5Zkyg?I zUCYWXN7_h$w37}JB%P#-gh)3DlO7Twy`+!ylL6?>bNL*FZahg1Lf;-r4MD#~rG{aL z6H+6vGcQu3uxndVW3Vspq{d<2ze~knH@~GO;N;^{aX9scR07W4o75znlZez5oaYOv zBtKj?B{dB d$ sub-regions. In a BFS the depth-$< d$ region (the BFS -ball of radius $d - 1$) is connected, so at most one boundary -component of $R_{C'}$ can lie on the source side; (R2) is therefore -equivalent to ``the closure of the depth-$> d$ region adjacent to -$R_{C'}$ has at most one connected component.'' Multi-hole topology -arises when several disjoint depth-$> d$ ``lobes'' of $G$ sit inside -the same depth-$d$ component. +Multi-hole topology (where $R_{C'}$ encloses several disjoint +depth-$>d$ sub-regions) does \emph{not} require an additional +hypothesis: the inner outerplanar graph $O = G[V_{C'} \cap L_{d+1}]$ +captures the multi-hole structure as a disconnected (or +non-$2$-connected) outerplanar graph, and its outer-face boundary +closed walk serves as $B_{\mathrm{in}}$. In particular, two disjoint +$L_{d+1}$ triangles inside $R_{C'}$ joined by a bridge edge in $O$ +give a $B_{\mathrm{in}}$ that traverses the bridge twice and visits +the bridge endpoints twice each --- not a simple cycle, but a +well-defined closed walk on $V(O)$. In the special case $d = 0$ with single-vertex source $S = \{v_0\}$ -both hypotheses hold automatically: $R_{C'}$ is the star of $v_0$, -a topological closed disk with one boundary cycle (the link of $v_0$), -giving a tire graph with degenerate inner boundary $\{v_0\}$. +(R1) holds automatically: $R_{C'}$ is the star of $v_0$, a topological +closed disk with one boundary cycle (the link of $v_0$), giving a +tire graph with degenerate outer boundary $\{v_0\}$. \end{remark} \begin{thebibliography}{9}