From db245eecea584c444be97397a9a24dd6676bc45a Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 20 May 2026 01:20:27 -0400 Subject: [PATCH] Prove outerplanarity and draft edge-flip resolution algorithm - Promote Prop 3.1 (outerplanarity of level subgraphs) to Theorem 3.1 with a proof by contradiction via a BFS-path argument; drop the $n \leq 10$ caveat and the now-resolved open question. - Add Section 5 "An edge-flip resolution algorithm": apex classification of $L_k$-edges, bridge lemma, cross-level flip pass, definition of tricky-everywhere odd cycles and facial depth (seeded from inner faces with $\geq 2$ outer-face edges), and the depth-guided flip procedure. Observation 5.5 records empirical termination at $n = 9, 10, 11$; Question 5.6 asks if it holds in general. - Add experiments/depth_monovariant_check.py (sanity check over triangulation iso-classes, confirms the count-of-tricky-faces monovariant strictly decreases per flip on all 1400 tricky configs at $n \leq 11$), viz_cycling.py and debug_cycling.py, and cycling_visualization.png illustrating the depth-definition fix. Co-Authored-By: Claude Opus 4.7 --- .../cycling_visualization.png | Bin 0 -> 310288 bytes .../experiments/debug_cycling.py | 88 +++++ .../experiments/depth_monovariant_check.py | 369 ++++++++++++++++++ .../experiments/viz_cycling.py | 197 ++++++++++ .../paper.aux | 33 +- .../paper.log | 221 +++++++---- .../paper.out | 11 +- .../paper.pdf | Bin 235833 -> 254719 bytes .../paper.tex | 185 +++++++-- 9 files changed, 992 insertions(+), 112 deletions(-) create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/cycling_visualization.png create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/debug_cycling.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/depth_monovariant_check.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/viz_cycling.py diff --git a/papers/level_resolutions_of_maximal_planar_graphs/cycling_visualization.png 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zS6S96MqY6rmy~IPz(t2U1x^yQZk(8)(p~>=HyS?GDlrm961b=5s3xlh=A+C-Lj;2F zZ`(<$7Zoe_vBn167nb1?k@pyG2ocTkkN2p7lO|*dz5Q9%?Vp1;u&s9cCIrTy(j+&1 zWoWXB^MdYODs$C<_adEtyW+%T`{*1+ZaumX?%4(}*s%sI3AwL@#xfH}O66V%rSew2 zt}fUtcz1c0yO9TRk=#*3!`yLz?};~S!IX_ykJ2zj54b|xbNCmQJtJ)Kc~PHQo4BN` zjKyW&ykD2%YP}VQQ)fdaTSA3Nq@%hUlLD5Gx5Vca@UpXWfzKYEuGcE_=9RK0@q+p7 zV~0i*!8`%Cf=;(zpYV-x5EA_Qc(B&upVvEB6mmZ=k$=Bt^~Jy4^1~$lHHUxA;g@sx gf8!mFxXm&pe9tx ({w},{x})") + ip2, emb_p = nx.check_planarity(Gp) + dump(Gp, emb_p, levels, nodes_k, 1, ('face', (0, 4, 2))) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/depth_monovariant_check.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/depth_monovariant_check.py new file mode 100644 index 0000000..b58574f --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/depth_monovariant_check.py @@ -0,0 +1,369 @@ +""" +Sanity check for the facial-depth monovariant on tricky (k,k) configurations. + +For each (G, S, k) where L_k has at least one "tricky-everywhere" odd face +(every edge is a (k,k) apex pair): + - Run the proposed algorithm: pick the deepest tricky-everywhere odd face, + flip the edge whose other-side face has min facial depth. + - Iterate until no tricky-everywhere odd face remains, or step budget hits. + - Track multiple candidate monovariants per flip. + +Reports counts of (a) terminations within budget, (b) cycle/failure cases, +(c) which monovariant (if any) decreases on every flip. +""" +import sys, os +sys.path.insert(0, os.path.dirname(os.path.abspath(__file__))) + +import networkx as nx +from collections import defaultdict, deque +from triangulation_gen import enumerate_all_triangulations +from level_cycles import ( + compute_levels, get_all_faces, inherited_embedding, level_sources +) + + +# -- helpers ------------------------------------------------------------ + +def faces_of_subgraph(G, emb_G, nodes_k): + if len(nodes_k) < 3: + return [] + L_k = G.subgraph(nodes_k) + if L_k.number_of_edges() < 3: + return [] + try: + emb_L = inherited_embedding(emb_G, nodes_k) + return get_all_faces(emb_L) + except Exception: + ip, emb_L = nx.check_planarity(L_k) + if not ip: + return [] + return get_all_faces(emb_L) + + +def canonical_face(face): + rots = [tuple(face[i:] + face[:i]) for i in range(len(face))] \ + + [tuple(face[::-1][i:] + face[::-1][:i]) for i in range(len(face))] + return min(rots) + + +def edges_of_face(face): + n = len(face) + return [(face[i], face[(i+1) % n]) for i in range(n)] + + +def apex_levels_for_edge(G, emb_G, u, v, levels): + if not G.has_edge(u, v): + return None + f1 = emb_G.traverse_face(u, v) + f2 = emb_G.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: + return None + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + return (levels.get(w), levels.get(x)), (w, x) + + +def identify_outer_face(faces, G, emb_G, levels, k): + """Outer face = the face whose 'outside' connects to lower levels. + Score each face by how many edges have at least one apex at level < k + on the opposite side; pick highest score, longest boundary as tiebreak.""" + best = (-1, -1, None) + for face in faces: + score = 0 + for (u, v) in edges_of_face(face): + if not G.has_edge(u, v): + continue + f1 = emb_G.traverse_face(u, v) + f2 = emb_G.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: + continue + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + if (levels.get(w, k) < k) or (levels.get(x, k) < k): + score += 1 + key = (score, len(face)) + if key > (best[0], best[1]): + best = (score, len(face), face) + return canonical_face(best[2]) if best[2] is not None else None + + +def face_depths(faces, outer_canon): + """BFS depth in the dual graph from the seed set: + inner faces with >= 2 edges incident to the outer face.""" + edge_to_faces = defaultdict(list) + canon_set = set() + for face in faces: + cf = canonical_face(face) + canon_set.add(cf) + for (a, b) in edges_of_face(face): + edge_to_faces[frozenset([a, b])].append(cf) + # Outer-face edges = edges of the outer face. + outer_face_edges = set() + for face in faces: + if canonical_face(face) == outer_canon: + for (a, b) in edges_of_face(face): + outer_face_edges.add(frozenset([a, b])) + break + # Seed set: inner faces with >= 2 outer-face edges. + seeds = [] + for face in faces: + cf = canonical_face(face) + if cf == outer_canon: + continue + count = 0 + for (a, b) in edges_of_face(face): + if frozenset([a, b]) in outer_face_edges: + count += 1 + if count >= 2: + seeds.append(cf) + if not seeds: + # Fall back: no seeds means no "easy" odd face nearby. Return empty. + return {} + D = nx.Graph() + for cf in canon_set: + if cf != outer_canon: + D.add_node(cf) + for e, fs in edge_to_faces.items(): + if len(fs) == 2: + f1, f2 = fs + if f1 != outer_canon and f2 != outer_canon: + D.add_edge(f1, f2) + # Multi-source BFS. + depths = {} + queue = deque() + for s in seeds: + if s in D: + depths[s] = 0 + queue.append(s) + while queue: + u = queue.popleft() + for v in D.neighbors(u): + if v not in depths: + depths[v] = depths[u] + 1 + queue.append(v) + return depths + + +def is_both_k_everywhere(face, G, emb_G, levels, k): + for (u, v) in edges_of_face(face): + result = apex_levels_for_edge(G, emb_G, u, v, levels) + if result is None: + return False + (la, lb), _ = result + if (la, lb) != (k, k): + return False + return True + + +def tricky_odd_faces(faces, G, emb_G, levels, k, depths, outer_canon): + """Return list of (canon, face, depth) for odd faces that are tricky- + everywhere AND have at least one (k,k) edge whose other-side face is + not the outer face (i.e., we can flip toward an inner neighbor).""" + out = [] + for face in faces: + cf = canonical_face(face) + if cf == outer_canon: + continue + if len(face) % 2 != 1: + continue + if not is_both_k_everywhere(face, G, emb_G, levels, k): + continue + d = depths.get(cf) + if d is None: + continue + out.append((cf, face, d)) + return out + + +def odd_faces_all(faces, depths, outer_canon): + out = [] + for face in faces: + cf = canonical_face(face) + if cf == outer_canon: + continue + if len(face) % 2 != 1: + continue + d = depths.get(cf) + if d is None: + continue + out.append((cf, face, d)) + return out + + +def lowest_depth_neighbor_flip(face, G, emb_G, levels, k, depths, faces, + outer_canon): + """Among (k,k) edges of `face`, return the flip toward the lowest-depth + non-outer neighbor face. Returns (u, v, w, x, neighbor_depth) or None.""" + edge_to_other = {} + face_canon = canonical_face(face) + for f in faces: + cf = canonical_face(f) + if cf == face_canon: + continue + for (a, b) in edges_of_face(f): + e = frozenset([a, b]) + edge_to_other.setdefault(e, cf) + best = None + for (u, v) in edges_of_face(face): + result = apex_levels_for_edge(G, emb_G, u, v, levels) + if result is None: + continue + (la, lb), (w, x) = result + if (la, lb) != (k, k): + continue + if G.has_edge(w, x): + continue + e = frozenset([u, v]) + other_canon = edge_to_other.get(e) + if other_canon is None or other_canon == outer_canon: + continue + other_depth = depths.get(other_canon) + if other_depth is None: + continue + if best is None or other_depth < best[4]: + best = (u, v, w, x, other_depth) + return best + + +# -- main routine ------------------------------------------------------- + +def analyze_config(G, emb, levels, nodes_k, k, max_steps=20): + """Run the user's algorithm iteratively. Returns: + {'terminated': bool, 'steps': n, 'trace': list of monovariant tuples} + A trace entry per step is (max_tricky_depth, num_tricky, max_odd_depth). + """ + Gc = G.copy() + emb_c = emb + trace = [] + for step in range(max_steps): + faces = faces_of_subgraph(Gc, emb_c, nodes_k) + if len(faces) < 2: + return {'terminated': True, 'steps': step, 'trace': trace, + 'reason': 'no_faces'} + outer = identify_outer_face(faces, Gc, emb_c, levels, k) + depths = face_depths(faces, outer) + if depths is None: + return {'terminated': True, 'steps': step, 'trace': trace, + 'reason': 'no_depths'} + odd_all = odd_faces_all(faces, depths, outer) + tricky = tricky_odd_faces(faces, Gc, emb_c, levels, k, depths, outer) + max_tricky = max((d for _, _, d in tricky), default=-1) + max_odd = max((d for _, _, d in odd_all), default=-1) + trace.append((max_tricky, len(tricky), max_odd)) + if not tricky: + return {'terminated': True, 'steps': step, 'trace': trace, + 'reason': 'no_tricky'} + # Pick deepest tricky face; flip toward lowest-depth neighbor. + cf_t, face_t, d_t = max(tricky, key=lambda x: x[2]) + flip = lowest_depth_neighbor_flip(face_t, Gc, emb_c, levels, k, + depths, faces, outer) + if flip is None: + return {'terminated': False, 'steps': step, 'trace': trace, + 'reason': 'no_flip'} + u, v, w, x, _ = flip + Gc.remove_edge(u, v) + Gc.add_edge(w, x) + ip, emb_c = nx.check_planarity(Gc) + if not ip: + return {'terminated': False, 'steps': step, 'trace': trace, + 'reason': 'nonplanar'} + return {'terminated': False, 'steps': max_steps, 'trace': trace, + 'reason': 'budget'} + + +def monovariant_decreases(trace, key): + """Does `trace[i][key]` strictly decrease at each step (except when + the algorithm has already terminated)?""" + vals = [t[key] for t in trace] + for i in range(1, len(vals)): + if vals[i] >= vals[i - 1]: + return False + return True + + +def run_check(n_values, max_steps=20): + total_configs = 0 + terminated = 0 + nonterm = 0 + mono_max_tricky = 0 + mono_count_tricky = 0 + mono_max_odd = 0 + nonterm_examples = [] + + for n in n_values: + print(f"\n=== n = {n} ===") + tris = enumerate_all_triangulations(n) + print(f" {len(tris)} iso-classes") + for tri_idx, G in enumerate(tris): + ip, emb = nx.check_planarity(G) + if not ip: + continue + for kind, label, source_set in level_sources(G, emb): + levels = compute_levels(G, source_set) + by_level = defaultdict(list) + for v, lv in levels.items(): + by_level[lv].append(v) + for k, nodes_k in by_level.items(): + if k == 0: + continue + faces = faces_of_subgraph(G, emb, nodes_k) + if len(faces) < 2: + continue + outer = identify_outer_face(faces, G, emb, levels, k) + if outer is None: + continue + depths = face_depths(faces, outer) + if depths is None: + continue + tricky = tricky_odd_faces(faces, G, emb, levels, k, + depths, outer) + if not tricky: + continue + total_configs += 1 + result = analyze_config(G, emb, levels, nodes_k, k, + max_steps) + if result['terminated']: + terminated += 1 + else: + nonterm += 1 + if len(nonterm_examples) < 5: + nonterm_examples.append({ + 'n': n, 'tri_idx': tri_idx, + 'source': (kind, label), + 'k': k, + 'reason': result['reason'], + 'trace': result['trace'], + 'steps': result['steps'], + }) + # Monovariant check (only meaningful if trace has >= 2) + tr = result['trace'] + if len(tr) >= 2: + if monovariant_decreases(tr, 0): + mono_max_tricky += 1 + if monovariant_decreases(tr, 1): + mono_count_tricky += 1 + if monovariant_decreases(tr, 2): + mono_max_odd += 1 + + print(f"\n=== summary ===") + print(f"tricky configs encountered: {total_configs}") + print(f" terminated within {max_steps} steps: {terminated}") + print(f" non-terminating / failed: {nonterm}") + print(f" candidate monovariants (strict decrease per step):") + print(f" max depth of tricky faces: {mono_max_tricky}") + print(f" count of tricky faces: {mono_count_tricky}") + print(f" max depth of all odd faces: {mono_max_odd}") + if nonterm_examples: + print(f"\nnon-terminating examples (first {len(nonterm_examples)}):") + for ex in nonterm_examples: + print(f" n={ex['n']} tri={ex['tri_idx']} src={ex['source']} " + f"k={ex['k']} reason={ex['reason']} steps={ex['steps']}") + print(f" trace (max_tricky, n_tricky, max_odd): {ex['trace']}") + + +if __name__ == "__main__": + if len(sys.argv) > 1: + ns = [int(x) for x in sys.argv[1:]] + else: + ns = [9, 10] + run_check(ns) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/viz_cycling.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/viz_cycling.py new file mode 100644 index 0000000..b63e8d9 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/viz_cycling.py @@ -0,0 +1,197 @@ +"""Visualize the cycling case for the (k,k) tricky-everywhere algorithm. + +Shows L_1 of triangulation index 6 at n=10, with source face (0,4,2). +Renders 4 steps of the algorithm side by side. +""" +import sys, os +sys.path.insert(0, os.path.dirname(os.path.abspath(__file__))) +import networkx as nx +import numpy as np +import matplotlib.pyplot as plt +import matplotlib.patches as mpatches +from collections import defaultdict +from triangulation_gen import enumerate_all_triangulations +from level_cycles import compute_levels +from depth_monovariant_check import ( + faces_of_subgraph, canonical_face, edges_of_face, apex_levels_for_edge, + identify_outer_face, face_depths, is_both_k_everywhere, + tricky_odd_faces, lowest_depth_neighbor_flip, +) + + +def layout_outerplanar(nodes_k, outer_face, G_full): + """Place outer-face vertices on a circle in the cyclic order they appear, + and interior vertices via a weighted average of their neighbors on the + circle.""" + outer_list = list(outer_face) + n = len(outer_list) + pos = {} + for i, v in enumerate(outer_list): + angle = np.pi / 2 - 2 * np.pi * i / n + pos[v] = np.array([np.cos(angle), np.sin(angle)]) + interior = [v for v in nodes_k if v not in pos] + if interior: + Lk = G_full.subgraph(nodes_k) + for v in interior: + nbrs = [u for u in Lk.neighbors(v) if u in pos] + if nbrs: + pos[v] = np.mean([pos[u] for u in nbrs], axis=0) + else: + pos[v] = np.array([0.0, 0.0]) + return pos + + +def draw_panel(ax, G_full, emb, levels, nodes_k, k, title, + highlight_edge=None, new_edge=None): + """Draw L_k with faces shaded by depth and tricky odd face highlighted.""" + faces = faces_of_subgraph(G_full, emb, nodes_k) + outer = identify_outer_face(faces, G_full, emb, levels, k) + depths = face_depths(faces, outer) + tricky = tricky_odd_faces(faces, G_full, emb, levels, k, depths, outer) + tricky_canons = {cf for cf, _, _ in tricky} + + # Find the outer face actual tuple (for layout) + outer_face = None + for f in faces: + if canonical_face(f) == outer: + outer_face = f + break + pos = layout_outerplanar(nodes_k, outer_face, G_full) + + # Shade faces by depth (skip outer face) + max_depth = max(d for d in depths.values()) if depths else 1 + for f in faces: + cf = canonical_face(f) + if cf == outer: + continue + d = depths.get(cf, 0) + if cf in tricky_canons: + color = '#ff5555' # red for tricky + alpha = 0.7 + else: + # Gradient from light yellow (depth 1) to orange (deep) + shade = 0.3 + 0.4 * (d / max(max_depth, 1)) + color = (1.0, 1.0 - shade * 0.3, 0.7 - shade * 0.4) + alpha = 0.5 + poly = plt.Polygon([pos[v] for v in f], color=color, alpha=alpha, + zorder=1) + ax.add_patch(poly) + # Label face center with depth + cx, cy = np.mean([pos[v] for v in f], axis=0) + face_len = len(f) + parity_mark = ' (odd)' if face_len % 2 == 1 else '' + tricky_mark = ' TRICKY' if cf in tricky_canons else '' + ax.text(cx, cy, f"d={d}\nlen={face_len}{parity_mark}{tricky_mark}", + fontsize=9, ha='center', va='center', zorder=3, + fontweight='bold' if cf in tricky_canons else 'normal') + + # Draw edges of L_k + Lk = G_full.subgraph(nodes_k) + hl_set = frozenset(highlight_edge) if highlight_edge else None + new_set = frozenset(new_edge) if new_edge else None + for u, v in Lk.edges(): + eset = frozenset([u, v]) + if hl_set and eset == hl_set: + ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]], + color='red', linewidth=4.0, zorder=2.5, + linestyle='--') + else: + ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]], + color='black', linewidth=1.6, zorder=2) + # Sketch where the new edge will land (dotted green) + if new_set: + nu, nv = new_edge + if nu in pos and nv in pos: + ax.plot([pos[nu][0], pos[nv][0]], [pos[nu][1], pos[nv][1]], + color='green', linewidth=2.5, zorder=2.5, + linestyle=':') + + # Draw vertices + for v in nodes_k: + ax.plot(pos[v][0], pos[v][1], 'o', color='black', markersize=12, + zorder=4) + ax.text(pos[v][0] * 1.13, pos[v][1] * 1.13, str(v), fontsize=14, + ha='center', va='center', color='blue', zorder=5, + fontweight='bold') + + ax.set_title(title, fontsize=12) + ax.set_aspect('equal') + ax.axis('off') + ax.set_xlim(-1.3, 1.3) + ax.set_ylim(-1.3, 1.3) + return tricky + + +def main(): + tris = enumerate_all_triangulations(10) + G = tris[6] + ip, emb = nx.check_planarity(G) + source = {0, 4, 2} + levels = compute_levels(G, source) + by_level = defaultdict(list) + for v, lv in levels.items(): + by_level[lv].append(v) + nodes_k = by_level[1] + k = 1 + + fig, axes_grid = plt.subplots(2, 2, figsize=(14, 14)) + axes = axes_grid.flatten() + fig.suptitle( + f"Cycling case: n=10, tri[6], source face (0,4,2), $L_{k}$\n" + f"Red = tricky-everywhere odd face (depth 2). Yellow = depth-1 odd " + f"faces (have free edges). Algorithm flips a (k,k) edge of the red " + f"face toward its lowest-depth neighbor each step.", + fontsize=13 + ) + + Gc = G.copy() + emb_c = emb + for step in range(4): + ax = axes[step] + # Compute flip choice first so we can render highlight before draw + faces = faces_of_subgraph(Gc, emb_c, nodes_k) + outer = identify_outer_face(faces, Gc, emb_c, levels, k) + depths = face_depths(faces, outer) + tricky_pre = tricky_odd_faces(faces, Gc, emb_c, levels, k, depths, + outer) + flip = None + if tricky_pre: + cf_t, face_t, d_t = max(tricky_pre, key=lambda x: x[2]) + flip = lowest_depth_neighbor_flip(face_t, Gc, emb_c, levels, k, + depths, faces, outer) + hl = None + new_e = None + if flip is not None: + u, v, w, x, _ = flip + hl = (u, v) + new_e = (w, x) + tricky = draw_panel(ax, Gc, emb_c, levels, nodes_k, k, + f"Step {step}", + highlight_edge=hl, new_edge=new_e) + if not tricky: + ax.set_title(f"Step {step}: no tricky face — done", + fontsize=12) + continue + if flip is None: + ax.set_title(f"Step {step}: no flip available", fontsize=12) + continue + u, v, w, x, _ = flip + ax.set_title( + f"Step {step}: tricky face {face_t}\n" + f"red-dashed: edge ({u},{v}) being flipped → " + f"green-dotted: new edge ({w},{x})", + fontsize=11) + # Apply flip + Gc.remove_edge(u, v) + Gc.add_edge(w, x) + ip2, emb_c = nx.check_planarity(Gc) + + plt.tight_layout(rect=[0, 0, 1, 0.93]) + out = os.path.join(os.path.dirname(os.path.abspath(__file__)), + '..', 'cycling_visualization.png') + plt.savefig(out, dpi=130, bbox_inches='tight') + print(f"saved: {out}") + + +if __name__ == "__main__": + main() diff --git a/papers/level_resolutions_of_maximal_planar_graphs/paper.aux b/papers/level_resolutions_of_maximal_planar_graphs/paper.aux index b166e5a..e483b73 100644 --- a/papers/level_resolutions_of_maximal_planar_graphs/paper.aux +++ b/papers/level_resolutions_of_maximal_planar_graphs/paper.aux @@ -19,26 +19,39 @@ \citation{appelhaken} \citation{rsst} \@writefile{toc}{\contentsline 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