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 <noreply@anthropic.com>

papers/level_resolutions_of_maximal_planar_graphs/experiments/debug_cycling.py

@@ -0,0 +1,88 @@
+"""Dump one cycling case to see what's going on."""
+import sys, os
+sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
+import networkx as nx
+from collections import defaultdict
+from triangulation_gen import enumerate_all_triangulations
+from level_cycles import compute_levels, level_sources
+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 dump(G, emb, levels, nodes_k, k, source_label):
+    print(f"  source: {source_label}, levels:")
+    by_level = defaultdict(list)
+    for v, lv in levels.items():
+        by_level[lv].append(v)
+    for lv in sorted(by_level):
+        print(f"    L_{lv} = {sorted(by_level[lv])}")
+    print(f"  examining L_{k} = {sorted(nodes_k)}")
+    L_k = G.subgraph(nodes_k)
+    print(f"  L_k edges: {sorted(L_k.edges())}")
+    faces = faces_of_subgraph(G, emb, nodes_k)
+    print(f"  faces of L_k (inherited embedding):")
+    for f in faces:
+        print(f"    {f}  len={len(f)}  canon={canonical_face(f)}")
+    outer = identify_outer_face(faces, G, emb, levels, k)
+    print(f"  outer face: {outer}")
+    depths = face_depths(faces, outer)
+    print(f"  depths: {depths}")
+    print(f"  edge-by-edge apex-level pairs:")
+    for f in faces:
+        cf = canonical_face(f)
+        if cf == outer:
+            continue
+        if len(f) < 3:
+            continue
+        print(f"    face {f} (depth={depths.get(cf)}, len={len(f)}):")
+        for (u, v) in edges_of_face(f):
+            result = apex_levels_for_edge(G, emb, u, v, levels)
+            if result is None:
+                print(f"      ({u},{v}): no apex")
+                continue
+            (la, lb), (w, x) = result
+            print(f"      ({u},{v}): apexes ({w}@L{la}, {x}@L{lb})")
+    tricky = tricky_odd_faces(faces, G, emb, levels, k, depths, outer)
+    print(f"  tricky odd faces: {[(f, d) for cf, f, d in tricky]}")
+    if tricky:
+        cf_t, face_t, d_t = max(tricky, key=lambda x: x[2])
+        print(f"  picking deepest tricky face: {face_t} at depth {d_t}")
+        flip = lowest_depth_neighbor_flip(face_t, G, emb, levels, k,
+                                           depths, faces, outer)
+        print(f"  flip choice: {flip}")
+
+
+# n=10, tri_idx=6, source face (0,4,2), k=1
+tris = enumerate_all_triangulations(10)
+G = tris[6]
+ip, emb = nx.check_planarity(G)
+print(f"G edges: {sorted(G.edges())}")
+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]
+dump(G, emb, levels, nodes_k, 1, ('face', (0, 4, 2)))
+
+print("\n--- after one flip ---")
+flip_choice = None
+faces = faces_of_subgraph(G, emb, nodes_k)
+outer = identify_outer_face(faces, G, emb, levels, 1)
+depths = face_depths(faces, outer)
+tricky = tricky_odd_faces(faces, G, emb, levels, 1, depths, outer)
+if tricky:
+    cf_t, face_t, d_t = max(tricky, key=lambda x: x[2])
+    flip_choice = lowest_depth_neighbor_flip(face_t, G, emb, levels, 1,
+                                              depths, faces, outer)
+if flip_choice:
+    u, v, w, x, _ = flip_choice
+    Gp = G.copy()
+    Gp.remove_edge(u, v)
+    Gp.add_edge(w, x)
+    print(f"Flipped ({u},{v}) -> ({w},{x})")
+    ip2, emb_p = nx.check_planarity(Gp)
+    dump(Gp, emb_p, levels, nodes_k, 1, ('face', (0, 4, 2)))

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)

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()

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-\OT1/cmr/m/n/10 Equiv-a-lently, ev-ery max-i-mal pla-nar graph (tri-an-gu-la-tion) is 4-colorable. The Appel--
+\OT1/cmr/m/n/10 Equiv-a-lently, ev-ery max-i-mal pla-nar graph (tri-an-gu-la-ti
+on) is 4-colorable. The Appel--
  []
 
 
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-\OT1/cmr/m/n/10 Haken proof [[]] and sub-se-quent Robertson--Sanders--Seymour--Thomas re-fine-ment [[]]
+\OT1/cmr/m/n/10 Haken proof [[]] and sub-se-quent Robertson--Sanders--Seymour--
+Thomas re-fine-ment [[]]
  []
 
 
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-\OT1/cmr/m/n/10 rely on dis-charg-ing ar-gu-ments and computer-verified re-ducible con-fig-u-ra-tions. Human-
+\OT1/cmr/m/n/10 rely on dis-charg-ing ar-gu-ments and computer-verified re-duci
+ble con-fig-u-ra-tions. Human-
  []
 
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papers/level_resolutions_of_maximal_planar_graphs/paper.out

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papers/level_resolutions_of_maximal_planar_graphs/paper.tex

@@ -88,8 +88,7 @@ source if the subgraphs of $G'$ induced by even- and odd-level vertices are
 both bipartite. By construction, any level resolution admits an explicit
 4-coloring obtained by 2-coloring each parity subgraph independently. The
 structural foundation of this approach is that each level subgraph $L_k$
-of $G$ is outerplanar (verified for all triangulations and sources at
+of $G$ is outerplanar, and outerplanar graphs are 3-chromatic; the level-resolution
-$n \leq 10$), and outerplanar graphs are 3-chromatic; the level-resolution
 problem is precisely to flip edges of $G$ to reduce each $L_k$ from
 chromatic number $3$ to $2$. We present computational results characterizing
 which isomorphism classes of maximal planar graphs on $n = 6, \ldots, 11$
@@ -179,20 +178,49 @@ disjoint color sets and so are properly colored.
 For each integer $k \geq 0$ and each $(G, S)$, write $L_k$ for the subgraph
 of $G$ induced by the level-$k$ vertices.
 
-\begin{proposition}
+\begin{theorem}
-\label{prop:outerplanar}
+\label{thm:outerplanar}
 For every plane triangulation $G$ and every level source $S$ of $G$, each
 level subgraph $L_k$ is outerplanar.
-\end{proposition}
+\end{theorem}
 
-A planar embedding witnessing outerplanarity is inherited from $G$: in the
+\begin{proof}
-planar embedding $\Pi_G$, the vertices at distance $\leq k - 1$ from the
+For $k = 0$, $L_0$ is either a single vertex (when $S$ is a degree-3
-source lie strictly on one side of the boundary of $L_k$, so all $L_k$
+vertex) or the triangle bounding the source face (when $S$ is a face),
-vertices can be placed on a common face of $L_k$. We have verified this
+both outerplanar. Fix $k \geq 1$ and suppose, for contradiction, that
-property computationally for every $(G, S)$ pair with $G$ on $n \leq 10$
+$L_k$ is not outerplanar.
-vertices ($14182$ pairs total, all yielding outerplanar level subgraphs).
 
-The combinatorial significance of Proposition~\ref{prop:outerplanar} is
+Let $D_k$ denote the planar drawing of $L_k$ inherited from $\Pi_G$:
+that is, the set of points and curves in the plane representing the
+vertices and edges of $L_k$ exactly as they appear in the embedding
+$\Pi_G$. Since $L_k$ is not outerplanar, no face of $D_k$ has every
+vertex of $L_k$ on its boundary.
+
+Let $F^\ast$ be the face of $D_k$ containing the source: when
+$S = \{v\}$, the face containing the point $v$; when $S$ is a face $F$
+of $G$, the face containing the open region of $F$ together with its
+three bounding vertices. The latter is well defined because each
+vertex of $F$ lies at level $0$ (hence is not a vertex of $L_k$) and
+each edge of $F$ joins two level-$0$ vertices (hence is not an edge of
+$L_k$), so $F$ and its boundary lie in a single component of
+$\mathbb{R}^2 \setminus D_k$. By assumption there exists $u \in L_k$
+with $u \notin \partial F^\ast$.
+
+Choose a BFS path $P : v_0, v_1, \ldots, v_k = u$ with $v_0 \in S$ and
+$v_i \in L_i$. For $0 \leq i \leq k - 1$, $v_i$ lies in $L_i$ and so is
+not a vertex of $L_k$; for $1 \leq i \leq k$, the edge $v_{i-1} v_i$
+joins $L_{i-1}$ to $L_i$ and so is not an edge of $L_k$. Hence, viewed
+as a curve in the plane, $P$ meets the drawing $D_k$ only at its
+endpoint $u$.
+
+The complement $\mathbb{R}^2 \setminus D_k$ is open, and $P \setminus
+\{u\}$ is its continuous image of a connected set, hence lies in a
+single face of $D_k$. Since $v_0 \in F^\ast$, in fact
+$P \setminus \{u\} \subseteq F^\ast$, so $u \in \overline{F^\ast}$ and
+therefore $u \in \partial F^\ast$, contradicting the choice of $u$.
+\end{proof}
+
+The combinatorial significance of Theorem~\ref{thm:outerplanar} is
 that outerplanar graphs are $3$-chromatic~\cite{chartrand}: their chromatic
 number is at most $3$. Hence each $L_k$ admits an independent 3-coloring,
 giving an immediate (but suboptimal) coloring of $G$ using at most
@@ -328,7 +356,116 @@ BFS-level parity cardinality on the same vertex set. Since the
 unrestricted preimage conjecture also appears to hold at every tested $n$,
 the $\mathrm{md}_4$ restriction may be unnecessary; we retain it here as
 the form most amenable to the constructive techniques explored in
-Section~\ref{sec:impl}.
+Section~\ref{sec:flip-algorithm}.
+
+\section{An edge-flip resolution algorithm}
+\label{sec:flip-algorithm}
+
+We describe an iterative edge-flip procedure aimed at producing, for a given
+$(G, S)$, a triangulation $G'$ on the same vertex set whose simple level
+cycles (with respect to the $G$-levels from $S$) are all even.
+
+\subsection{Apex classification of $L_k$-edges}
+
+Let $k \geq 1$. For each $uv \in E(L_k)$, the two triangles of $G$ bounding
+$uv$ have third vertices $w, x$, called the \emph{apexes} of $uv$, with
+$\ell_G(w), \ell_G(x) \in \{k-1, k, k+1\}$ by BFS. We call $uv$
+\emph{intra-level} when $\ell_G(w) = \ell_G(x) = k$, and \emph{cross-level}
+otherwise.
+
+\begin{lemma}
+\label{lem:bridge-apex}
+If both apexes of $uv \in E(L_k)$ are at level $k - 1$, then $uv$ is a
+bridge of $L_k$.
+\end{lemma}
+
+\begin{proof}[Proof sketch]
+In a plane triangulation, the neighbors of $u$ in $G$ at level $\leq k - 1$
+form a contiguous arc in the cyclic order around $u$. If both apexes $w, x$
+of $uv$ lie at level $k - 1$ on opposite sides of $uv$, then $v$ lies in the
+complementary cyclic arc, which contains no other level-$k$ neighbor of $u$.
+The symmetric statement around $v$ gives that $u$ is $v$'s only level-$k$
+neighbor in the corresponding arc, so $uv$ is a bridge of $L_k$.
+\end{proof}
+
+In particular every edge on a cycle of $L_k$ has at least one apex at level
+$k$ or $k+1$.
+
+\begin{proposition}
+\label{prop:flip-target}
+Flipping $uv \in E(L_k)$ with apexes $w, x$ replaces $uv$ with $wx$ in $G$.
+The new edge $wx$ belongs to $L_k$ iff $\ell_G(w) = \ell_G(x) = k$, and to
+$L_{k+1}$ iff $\ell_G(w) = \ell_G(x) = k + 1$; otherwise $wx$ is cross-parity
+and lies in no level subgraph. In all cases $uv$ is removed from $L_k$.
+\end{proposition}
+
+\subsection{Cross-level flip pass}
+
+For each odd simple cycle $C$ of each $L_k$ containing a cross-level edge,
+flip one such edge. By Proposition~\ref{prop:flip-target} the new edge
+either enters $L_{k+1}$ (in the apex case $(k+1, k+1)$) or is cross-parity
+(otherwise). Choosing apex pairs distinctly across cycles makes the set of
+new edges entering any single level $L_j$ a matching, hence a forest, and
+similarly for new same-parity-distance-$2$ edges entering the relevant
+parity subgraph; these therefore introduce no odd cycle.
+
+\subsection{Tricky-everywhere cycles}
+
+After the cross-level pass, the only odd simple cycles remaining in any
+$L_k$ are those whose every edge is intra-level; we call such a cycle
+\emph{tricky-everywhere}. By Proposition~\ref{prop:flip-target}, flipping
+any edge of a tricky-everywhere cycle replaces it with another edge of $L_k$,
+so the local triangle pair $(uvw, uvx)$ becomes $(uwx, vwx)$: still a pair
+of odd triangles inside $L_k$. To make global progress on these cycles we
+use a facial-depth potential to choose the flip.
+
+\begin{definition}[Facial depth]
+\label{def:facial-depth}
+Let $L_k$ be drawn with the outerplanar embedding inherited from $\Pi_G$,
+let $D$ be the dual graph of this drawing with the outer face removed, and
+let $\mathcal{B}$ be the set of inner faces incident to at least two edges
+of the outer face of $L_k$. The \emph{facial depth} of an inner face $F$ of
+$L_k$ is
+\[
+  \mathrm{depth}(F) \;=\; \min_{F' \in \mathcal{B}} \mathrm{dist}_D(F, F'),
+\]
+with the convention $\mathrm{depth}(F) = \infty$ if no such $F'$ exists.
+\end{definition}
+
+\subsection{The algorithm}
+
+\begin{enumerate}
+\item \emph{Cross-level flip pass.} For each $L_k$ and each odd simple cycle
+$C \subseteq L_k$ containing a cross-level edge, flip one such edge,
+selecting apex pairs to keep the newly added edges a matching in each
+target level subgraph and in the relevant parity subgraph.
+\item \emph{Intra-level flip loop.} While some $L_k$ contains a
+tricky-everywhere odd simple cycle:
+\begin{enumerate}
+\item compute facial depths for all simple level cycles of $L_k$;
+\item among tricky-everywhere odd simple cycles of maximum facial depth,
+      pick any $C$;
+\item among the edges of $C$, pick one whose other incident inner face has
+      minimum facial depth, and flip it.
+\end{enumerate}
+\end{enumerate}
+
+\begin{observation}
+\label{obs:terminate}
+For every plane triangulation $G$ on $n \in \{9, 10, 11\}$ vertices, every
+level source $S$, and every $k$ such that $L_k$ contains a tricky-everywhere
+odd simple cycle, Step~2 terminates with no tricky-everywhere odd simple
+cycle remaining in any $L_k$. Moreover, the total number of
+tricky-everywhere odd simple cycles strictly decreases on every flip chosen
+by Step~2(c).
+\end{observation}
+
+\begin{question}
+\label{q:terminate-all-n}
+Does Observation~\ref{obs:terminate} hold for all $n$? Equivalently, does
+the count of tricky-everywhere odd simple cycles strictly decrease on every
+Step~2(c) flip, in every plane triangulation?
+\end{question}
 
 \section{Discussion and open questions}
 
@@ -340,7 +477,7 @@ The computational results suggest the following:
       arise as level resolutions, and the icosahedron does at $n = 12$
       (Observations~\ref{obs:preimage} and~\ref{obs:icosa}).
 \item Each level subgraph $L_k$ of $G$ is outerplanar
-      (Proposition~\ref{prop:outerplanar}), so each $L_k$ is 3-chromatic
+      (Theorem~\ref{thm:outerplanar}), so each $L_k$ is 3-chromatic
       classically and independently of 4CT. The level-resolution problem
       reduces to flipping edges of $G$ so that each $L_k$'s chromatic
       number drops from $3$ to $2$, while avoiding creation of $G$-level-2
@@ -355,20 +492,12 @@ The computational results suggest the following:
       that does not invoke 4CT.
 \end{enumerate}
 
-\begin{question}
+The algorithm of Section~\ref{sec:flip-algorithm} is the candidate
-Given that each $L_k$ is outerplanar, can the odd cycles of each $L_k$ in
+constructive answer: cross-level flips dispose of every odd cycle of $L_k$
-$G$ be broken by a globally consistent choice of flips? Equivalently: is
+that admits one, and the facial-depth-guided intra-level flip loop attacks
-there a constructive procedure that, starting from $G$ with source $S$,
+the residual tricky-everywhere cycles. Observation~\ref{obs:terminate}
-produces $G'$ such that each $L_k$ is bipartite in $G'$ and no $G$-level-2
+records that the loop terminates on all tested $(G, S, k)$ at $n \leq 11$;
-same-parity edges are introduced?
+Question~\ref{q:terminate-all-n} asks whether termination holds for all $n$.
-\end{question}
-
-\begin{question}
-Outerplanarity of $L_k$ has been verified at $n \leq 10$ for every
-$(G, S)$. Does it hold for all $n$? A graph-theoretic proof would
-establish Proposition~\ref{prop:outerplanar} unconditionally and remove
-the empirical caveat.
-\end{question}
 
 \section{Implementation}
 \label{sec:impl}