Handle compound medial tires in cut labelling

papers/medial_tire_cuts/experiments/medial_tire_dual_cut_experiment.py

@@ -57,12 +57,11 @@ sys.path.insert(0, _MTD)
 sys.path.insert(0, _CUT_LIB)
 
 from tire_realization_analysis import (  # noqa: E402
-    ekey, extract_tread, medial_graph, medial_tire_facemodel,
+    ekey, medial_graph, triangular_faces,
-    recognise, triangular_faces,
 )
 from run_medial_tire_cut_experiment import (  # noqa: E402
-    _assemble_cut_graph, _cap_cut, _label_treads,
+    _assemble_cut_graph, _build_treads, _cap_cut, _label_treads,
-    random_maximal_planar_min_degree,
+    random_maximal_planar_5_connected,
 )
 from medial_tire_cut_labelling import up_apex_cuts  # noqa: E402
 
@@ -71,31 +70,6 @@ from medial_tire_cut_labelling import up_apex_cuts  # noqa: E402
 # Tread recognition and the source-dual graph.
 # --------------------------------------------------------------------------- #
 
-def _build_treads(faces, levels):
-    """Recognise the full medial tire graph(s) of every BFS-level tread.
-
-    A tread depth whose annular frontier splits into several disjoint cycles
-    yields one tire per cycle.  Returns ``(treads, skipped)`` where ``treads``
-    maps ``(depth, component)`` to the recognised ``(g, bij)`` and ``skipped``
-    lists ``(d, reason)`` for the depths that produced no tire.
-    """
-    treads, skipped = {}, []
-    for d in range(max(levels.values())):
-        tread = extract_tread(faces, levels, d)
-        if tread is None:
-            skipped.append((d, "no tread faces"))
-            continue
-        if len(tread["up"]) < 3:
-            skipped.append((d, f"only {len(tread['up'])} up teeth"))
-            continue
-        tires = recognise(medial_tire_facemodel(tread["tread_faces"]), tread)
-        if not tires:
-            skipped.append((d, "no annular cycle recognised as a tire"))
-            continue
-        for c, gb in enumerate(tires):
-            treads[(d, c)] = gb
-    return treads, skipped
-
 
 def root_entry_choices(G, source):
     """Edge indices of the root tread's up teeth -- the eligible entry teeth.
@@ -104,7 +78,7 @@ def root_entry_choices(G, source):
     """
     faces, _ = triangular_faces(G)
     levels = nx.single_source_shortest_path_length(G, source)
-    treads, _ = _build_treads(faces, levels)
+    treads, _skipped, _meta = _build_treads(faces, levels)
     if not treads:
         return []
     g, _bij = treads[min(treads)]
@@ -209,12 +183,28 @@ def annular_cut_edges(results, cap_cuts):
 def up_apex_cut_edges(results):
     """Primal edges whose dual edge the apex duplications remove: the apex
     medial vertex of every up tooth across all treads, except each tread's
-    entry tooth and any vertex that is the shared apex of two up teeth."""
+    entry tooth and any vertex that is the shared apex of two up teeth within
+    that recognised annular cycle.  In a compound component, an entry apex is
+    treated as an entry seam for every cycle sharing that apex.
+    """
+    entry_apexes = defaultdict(set)
+    for key in sorted(results):
+        rec = results[key]
+        compound = rec.get("compound", key)
+        entry_apexes[compound].add(rec["bij"][f"u{rec['entry_edge']}"])
+
     removed = set()
     for key in sorted(results):
-        g, bij = results[key]["g"], results[key]["bij"]
+        rec = results[key]
-        entry = results[key]["entry_edge"]
+        g, bij = rec["g"], rec["bij"]
-        removed.update(up_apex_cuts(g, entry, bij=bij).values())
+        entry = rec["entry_edge"]
+        compound = rec.get("compound", key)
+        removed.update(
+            up_apex_cuts(
+                g, entry, bij=bij,
+                shared_apexes=entry_apexes.get(compound, set()),
+            ).values()
+        )
     return removed
 
 
@@ -245,7 +235,7 @@ def medial_tire_dual_cut(G, source, entry_edge):
     faces, emb = triangular_faces(G)
     M = medial_graph(G)
     levels = nx.single_source_shortest_path_length(G, source)
-    treads, skipped = _build_treads(faces, levels)
+    treads, skipped, tread_meta = _build_treads(faces, levels)
     if not treads:
         raise ValueError(f"level source {source} induces no recognised tread")
 
@@ -256,7 +246,8 @@ def medial_tire_dual_cut(G, source, entry_edge):
             f"(choices: {sorted(g_root.up_edges)})")
 
     results = {}
-    _label_treads(treads, results, root_entry_edge=entry_edge)
+    _label_treads(
+        treads, results, root_entry_edge=entry_edge, tread_meta=tread_meta)
     cap_cuts = _cap_cut(G, emb, source, levels, results)
     cut_graph, labels, warnings = _assemble_cut_graph(M, results, cap_cuts=cap_cuts)
 
@@ -276,7 +267,8 @@ def medial_tire_dual_cut(G, source, entry_edge):
         "G": G, "M": M, "source": source, "entry_edge": entry_edge,
         "entry_medial_vertex": entry_medial,
         "faces": faces, "outer_face": 0,
-        "levels": levels, "treads": treads, "skipped": skipped,
+        "levels": levels, "treads": treads, "tread_meta": tread_meta,
+        "skipped": skipped,
         "results": results, "cap_cuts": cap_cuts, "cut_graph": cut_graph,
         "labels": labels, "warnings": warnings,
         "dual": dual, "removed_dual_edges": removed,
@@ -319,8 +311,11 @@ def dual_cut_random_face(G, rng=None):
     raise ValueError("no face's cap vertex induces a recognised root tread")
 
 
-def random_dual_cut(n=20, seed=0, rng=None, min_degree=5, flips=400, attempts=1000):
+def random_dual_cut(
-    """Find a random maximal planar graph of minimum degree ``min_degree``, then
+    n=20, seed=0, rng=None, min_connectivity=5, flips=400, attempts=1000,
+    min_degree=None,
+):
+    """Find a random maximal planar graph of node connectivity 5, then
     ``dual_cut_random_face``.
 
     ``seed`` drives the graph sample; ``rng`` (defaulting to ``Random(seed)``)
@@ -328,12 +323,16 @@ def random_dual_cut(n=20, seed=0, rng=None, min_degree=5, flips=400, attempts=10
     pipeline is reproducible from ``(n, seed)``.
     """
     rng = rng or random.Random(seed)
-    G, graph_seed = random_maximal_planar_min_degree(
+    if min_degree is not None:
-        n, seed, flips=flips, min_degree=min_degree, attempts=attempts)
+        min_connectivity = max(min_connectivity, min_degree)
+    G, graph_seed = random_maximal_planar_5_connected(
+        n, seed, flips=flips, min_connectivity=min_connectivity, attempts=attempts)
     result = dual_cut_random_face(G, rng=rng)
     result["graph_seed"] = graph_seed
     result["base_min_degree"] = min(dict(G.degree()).values())
+    result["base_connectivity"] = nx.node_connectivity(G)
     result["min_degree"] = min(dict(result["G"].degree()).values())
+    result["connectivity"] = nx.node_connectivity(result["G"])
     return result
 
 
@@ -353,7 +352,8 @@ def summary(result):
         f"(deep embedding of base n="
         f"{base.number_of_nodes() if base is not None else '?'}) "
         f"graph_seed={result.get('graph_seed', '?')} "
-        f"min_degree={result.get('min_degree', min(dict(G.degree()).values()))}",
+        f"base_connectivity={result.get('base_connectivity', '?')} "
+        f"deep_connectivity={result.get('connectivity', nx.node_connectivity(G))}",
         f"chosen face: {result.get('chosen_face', '?')}  "
         f"-> cap vertex x*={result.get('cap_vertex', result['source'])}",
         f"level source: cap vertex {result['source']}  "
@@ -828,7 +828,9 @@ def main():
         description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
     parser.add_argument("-n", type=int, default=20, help="number of vertices")
     parser.add_argument("--seed", type=int, default=0, help="graph sample seed")
-    parser.add_argument("--min-degree", type=int, default=5)
+    parser.add_argument("--min-connectivity", type=int, default=5)
+    parser.add_argument("--min-degree", type=int, default=None,
+                        help="compatibility alias; also raises min-connectivity")
     parser.add_argument("--face", type=str, default=None,
                         help="fix the chosen face as 'a,b,c' (default: random "
                              "via rng); the deep embedding's cap vertex is the "
@@ -846,8 +848,11 @@ def main():
 
     rng = random.Random(args.seed)
     if args.face is not None:
-        G, graph_seed = random_maximal_planar_min_degree(
+        min_connectivity = args.min_connectivity
-            args.n, args.seed, min_degree=args.min_degree)
+        if args.min_degree is not None:
+            min_connectivity = max(min_connectivity, args.min_degree)
+        G, graph_seed = random_maximal_planar_5_connected(
+            args.n, args.seed, min_connectivity=min_connectivity)
         face = tuple(int(x) for x in args.face.split(","))
         G_prime, cap, depth = deep_embedding(G, face)
         if args.entry is not None:
@@ -860,10 +865,14 @@ def main():
         result["deep_depth"] = depth
         result["graph_seed"] = graph_seed
         result["base_min_degree"] = min(dict(G.degree()).values())
+        result["base_connectivity"] = nx.node_connectivity(G)
         result["min_degree"] = min(dict(G_prime.degree()).values())
+        result["connectivity"] = nx.node_connectivity(G_prime)
     else:
         result = random_dual_cut(n=args.n, seed=args.seed,
-                                 rng=rng, min_degree=args.min_degree)
+                                 rng=rng,
+                                 min_connectivity=args.min_connectivity,
+                                 min_degree=args.min_degree)
 
     print(summary(result))
     if args.png:

papers/medial_tire_cuts/experiments/run_medial_tire_cut_experiment.py

@@ -2,10 +2,8 @@
 
 End-to-end experiment for the *Medial Tire Cuts* paper:
 
-  1. Generate a random maximal planar graph G on n vertices (stacked seed plus
+  1. Generate a 5-connected maximal planar graph G on n vertices, using
-     random diagonal flips; ``random_maximal_planar`` from the medial tire
+     ``plantri -c5`` when available and verifying node connectivity.
-     decompositions experiments), optionally rejecting samples below a requested
-     minimum degree.
   2. Build its medial graph M(G).
   3. Take the nested tire decomposition at one random vertex level source: the
      BFS-level treads, each realized as a FullMedialTireGraph.
@@ -40,6 +38,7 @@ import os
 import random
 import subprocess
 import sys
+from collections import defaultdict
 
 import networkx as nx
 
@@ -56,12 +55,125 @@ sys.path.insert(0, _MTD)
 sys.path.insert(0, _CUT_LIB)
 
 from tire_realization_analysis import (  # noqa: E402
-    ekey, extract_tread, medial_graph, medial_tire_facemodel,
+    ekey, medial_graph, medial_tire_facemodel,
     random_maximal_planar, recognise, triangular_faces,
 )
 from medial_tire_cut_labelling import door_bite, label_and_cut  # noqa: E402
 
 
+# --------------------------------------------------------------------------- #
+# Component-based tread recognition.
+# --------------------------------------------------------------------------- #
+
+def _edge_face_data(faces):
+    edge_faces = defaultdict(list)
+    for i, face in enumerate(faces):
+        for x, y in ((face[0], face[1]), (face[1], face[2]), (face[2], face[0])):
+            edge_faces[ekey(x, y)].append(i)
+    return edge_faces
+
+
+def _depth_components(faces, edge_faces, levels):
+    depths = [min(levels[v] for v in face) for face in faces]
+    dual_adj = defaultdict(set)
+    for incident in edge_faces.values():
+        for a in range(len(incident)):
+            for b in range(a + 1, len(incident)):
+                dual_adj[incident[a]].add(incident[b])
+                dual_adj[incident[b]].add(incident[a])
+
+    comps = []
+    seen = [False] * len(faces)
+    for start in range(len(faces)):
+        if seen[start]:
+            continue
+        depth = depths[start]
+        stack = [start]
+        comp = []
+        seen[start] = True
+        while stack:
+            face = stack.pop()
+            comp.append(face)
+            for other in dual_adj[face]:
+                if not seen[other] and depths[other] == depth:
+                    seen[other] = True
+                    stack.append(other)
+        comps.append((depth, tuple(sorted(comp))))
+    return comps
+
+
+def _tread_from_component(faces, levels, face_indices):
+    tread_faces = [faces[i] for i in face_indices]
+    if not tread_faces:
+        return None
+    depth = min(min(levels[v] for v in face) for face in tread_faces)
+    annular, up, down = set(), set(), set()
+    face_of_down = defaultdict(int)
+    for face in tread_faces:
+        for x, y in ((face[0], face[1]), (face[1], face[2]), (face[2], face[0])):
+            e = ekey(x, y)
+            lx, ly = levels[x], levels[y]
+            if {lx, ly} == {depth, depth + 1}:
+                annular.add(e)
+            elif lx == ly == depth:
+                up.add(e)
+            elif lx == ly == depth + 1:
+                down.add(e)
+                face_of_down[e] += 1
+    if len(annular) < 3:
+        return None
+    return {
+        "tread_faces": tread_faces,
+        "annular": annular,
+        "up": up,
+        "down": down,
+        "bites": {e for e in down if face_of_down[e] == 2},
+    }
+
+
+def _build_treads(faces, levels):
+    """Recognise simple cycles inside connected depth components.
+
+    The returned ``treads`` keeps the existing simple-tire interface used by
+    the labelling code.  ``tread_meta`` records the connected depth component
+    each simple cycle came from, so compound tires can be chained through
+    shared up apexes rather than seeded as unrelated roots.
+    """
+    treads, skipped, tread_meta = {}, [], {}
+    edge_faces = _edge_face_data(faces)
+    comps = sorted(_depth_components(faces, edge_faces, levels),
+                   key=lambda item: (item[0], item[1]))
+    component_count = defaultdict(int)
+    tire_count = defaultdict(int)
+    for depth, face_indices in comps:
+        component = component_count[depth]
+        component_count[depth] += 1
+        tread = _tread_from_component(faces, levels, face_indices)
+        if tread is None:
+            skipped.append(((depth, component), "no tread faces"))
+            continue
+        if len(tread["up"]) < 3:
+            skipped.append(((depth, component), f"only {len(tread['up'])} up teeth"))
+            continue
+        tires = recognise(medial_tire_facemodel(tread["tread_faces"]), tread)
+        if not tires:
+            skipped.append(((depth, component), "no annular cycle recognised as a tire"))
+            continue
+        compound = (depth, component)
+        cycle_count = len(tires)
+        for cycle, gb in enumerate(tires):
+            key = (depth, tire_count[depth])
+            tire_count[depth] += 1
+            treads[key] = gb
+            tread_meta[key] = {
+                "compound": compound,
+                "cycle": cycle,
+                "cycle_count": cycle_count,
+                "face_indices": face_indices,
+            }
+    return treads, skipped, tread_meta
+
+
 # --------------------------------------------------------------------------- #
 # 4. Walk-depth labelling and cut, chained down the tire tree.
 # --------------------------------------------------------------------------- #
@@ -71,7 +183,7 @@ def _apex_vertex(g, bij, edge):
     return bij[g.apex_of_edge(edge)]
 
 
-def _label_treads(treads, results, root_entry_edge=None):
+def _label_treads(treads, results, root_entry_edge=None, tread_meta=None):
     """Fill ``results[(d, c)]`` with the walk-depth labelling and cuts for every
     recognised tire ``c`` of every tread depth ``d``, chaining child entries to
     parent down teeth.
@@ -99,28 +211,85 @@ def _label_treads(treads, results, root_entry_edge=None):
                 if apex not in parent_down or value < parent_down[apex]:
                     parent_down[apex] = value
         has_parent = any(k[0] == d - 1 for k in treads)
-        for key in sorted(k for k in treads if k[0] == d):
+        pending = sorted(k for k in treads if k[0] == d)
-            g, bij = treads[key]
+        while pending:
-            if not has_parent:
+            progressed = False
-                if (key == (root_d, 0) and root_entry_edge is not None
+            deferred = []
-                        and root_entry_edge in g.up_edges):
+            use_sibling_entries = has_parent and not any(
-                    entry_edge, start_depth = root_entry_edge, 0
+                parent_down.keys() & {treads[key][1][f"u{m}"]
-                else:
+                                      for m in treads[key][0].up_edges}
-                    entry_edge, start_depth = g.up_edges[0], 0  # arbitrary entry
+                for key in pending
-            else:
+                if key not in results
+            )
+            for key in pending:
+                if key in results:
+                    continue
+                g, bij = treads[key]
                 child_up_apex = {bij[f"u{m}"]: m for m in g.up_edges}
-                best = None
+                entry = None
-                for apex, value in parent_down.items():
+                if not has_parent:
-                    if apex in child_up_apex and (best is None or value < best[1]):
+                    if (key == (root_d, 0) and root_entry_edge is not None
-                        best = (child_up_apex[apex], value)
+                            and root_entry_edge in g.up_edges):
-                if best is not None:  # chains to a parent down tooth
+                        entry = (root_entry_edge, 0)
-                    entry_edge, start_depth = best
+                    else:
-                else:                  # no shared apex (degenerate); root-style
+                        entry = (g.up_edges[0], 0)  # arbitrary entry
-                    entry_edge, start_depth = g.up_edges[0], 0
+                else:
+                    best = None
+                    for apex, value in parent_down.items():
+                        if apex in child_up_apex and (best is None or value < best[1]):
+                            best = (child_up_apex[apex], value)
+                    if best is not None:  # chains to a parent down tooth
+                        entry = best
+                    elif use_sibling_entries:
+                        compound = (
+                            tread_meta.get(key, {}).get("compound")
+                            if tread_meta is not None else None
+                        )
+                        sibling_best = None
+                        if compound is not None:
+                            for sk, srec in results.items():
+                                if sk[0] != d or srec.get("compound") != compound:
+                                    continue
+                                sg, sbij = srec["g"], srec["bij"]
+                                for edge in sg.up_edges:
+                                    apex = sbij[f"u{edge}"]
+                                    if apex not in child_up_apex:
+                                        continue
+                                    value = srec["depth"][edge] + 1
+                                    if (sibling_best is None
+                                            or value > sibling_best[1]):
+                                        sibling_best = (child_up_apex[apex], value)
+                        if sibling_best is not None:
+                            entry = sibling_best
+
+                if entry is None:
+                    deferred.append(key)
+                    continue
+                entry_edge, start_depth = entry
+                depth, cuts = label_and_cut(g, entry_edge, start_depth=start_depth)
+                results[key] = {"g": g, "bij": bij, "entry_edge": entry_edge,
+                                "start_depth": start_depth, "depth": depth,
+                                "cuts": cuts}
+                if tread_meta is not None and key in tread_meta:
+                    results[key].update(tread_meta[key])
+                progressed = True
+
+            if progressed:
+                pending = deferred
+                continue
+
+            # Degenerate component: no parent or labelled sibling gives an
+            # entry.  Seed it so any remaining sibling cycles can chain to it.
+            key = deferred[0]
+            g, bij = treads[key]
+            entry_edge, start_depth = g.up_edges[0], 0
             depth, cuts = label_and_cut(g, entry_edge, start_depth=start_depth)
             results[key] = {"g": g, "bij": bij, "entry_edge": entry_edge,
                             "start_depth": start_depth, "depth": depth,
                             "cuts": cuts}
+            if tread_meta is not None and key in tread_meta:
+                results[key].update(tread_meta[key])
+            pending = deferred[1:]
 
 
 def _cap_cut(G, emb, source, levels, results):
@@ -252,39 +421,46 @@ def _assemble_cut_graph(M, results, cap_cuts=None):
 # Driver.
 # --------------------------------------------------------------------------- #
 
-def random_maximal_planar_min_degree(n: int, seed: int, flips: int = 400,
+def random_maximal_planar_5_connected(n: int, seed: int, flips: int = 400,
-                                     min_degree: int = 0,
+                                      min_connectivity: int = 5,
-                                     attempts: int = 1000) -> tuple[nx.Graph, int]:
+                                      attempts: int = 1000) -> tuple[nx.Graph, int]:
-    """Generate a random maximal planar graph with minimum degree at least
+    """Generate a maximal planar graph with node connectivity at least
-    ``min_degree``.  The returned seed is the actual sample seed used."""
+    ``min_connectivity``.  The returned seed is the actual sample seed used."""
-    if min_degree <= 0:
+    if min_connectivity <= 0:
         return random_maximal_planar(n, seed, flips=flips), seed
 
-    if min_degree >= 5:
+    if min_connectivity >= 5:
         plantri = os.path.normpath(os.path.join(_HERE, "..", "..", "..",
                                                 "plantri", "plantri"))
         if os.path.exists(plantri):
             data = subprocess.check_output(
-                [plantri, f"-m{min_degree}", "-g", str(n)],
+                [plantri, "-c5", "-g", str(n)],
                 stderr=subprocess.DEVNULL)
-            graphs = [line for line in data.splitlines() if line]
+            graphs = [
+                line for line in data.splitlines()
+                if line and not line.startswith(b">>")
+            ]
             if graphs:
-                G = nx.from_graph6_bytes(graphs[seed % len(graphs)])
+                for offset in range(len(graphs)):
-                return nx.convert_node_labels_to_integers(G), seed
+                    G = nx.from_graph6_bytes(graphs[(seed + offset) % len(graphs)])
+                    G = nx.convert_node_labels_to_integers(G)
+                    if nx.node_connectivity(G) >= min_connectivity:
+                        return G, seed + offset
 
     for offset in range(attempts):
         sample_seed = seed + offset
         G = random_maximal_planar(n, sample_seed, flips=flips)
-        if min(dict(G.degree()).values()) >= min_degree:
+        if nx.node_connectivity(G) >= min_connectivity:
             return G, sample_seed
     raise RuntimeError(
         f"no random maximal planar graph on {n} vertices with "
-        f"minimum degree >= {min_degree} found in {attempts} attempts "
+        f"node connectivity >= {min_connectivity} found in {attempts} attempts "
         f"starting at seed {seed}")
 
 
 def run_experiment(n: int = 12, seed: int = 0, flips: int = 400,
-                   min_degree: int = 5, attempts: int = 1000) -> dict:
+                   min_connectivity: int = 5, attempts: int = 1000,
+                   min_degree: int | None = None) -> dict:
     """Run the full pipeline and return a structured result.
 
     Result keys: ``n, seed, G, M, source, treads`` (dict depth -> (g, bij)),
@@ -292,38 +468,28 @@ def run_experiment(n: int = 12, seed: int = 0, flips: int = 400,
     (depth, reason)), ``cut_graph`` (networkx graph), ``labels`` (list of tooth
     records), ``warnings``.
     """
-    G, graph_seed = random_maximal_planar_min_degree(
+    if min_degree is not None:
-        n, seed, flips=flips, min_degree=min_degree, attempts=attempts)
+        min_connectivity = max(min_connectivity, min_degree)
+    G, graph_seed = random_maximal_planar_5_connected(
+        n, seed, flips=flips, min_connectivity=min_connectivity, attempts=attempts)
     faces, emb = triangular_faces(G)
     M = medial_graph(G)
     source = random.Random(f"source-{graph_seed}").choice(sorted(G.nodes()))
     levels = nx.single_source_shortest_path_length(G, source)
 
-    treads, skipped = {}, []
+    treads, skipped, tread_meta = _build_treads(faces, levels)
-    for d in range(max(levels.values())):
-        tread = extract_tread(faces, levels, d)
-        if tread is None:
-            skipped.append((d, "no tread faces"))
-            continue
-        if len(tread["up"]) < 3:
-            skipped.append((d, f"only {len(tread['up'])} up teeth"))
-            continue
-        tires = recognise(medial_tire_facemodel(tread["tread_faces"]), tread)
-        if not tires:
-            skipped.append((d, "no annular cycle recognised as a tire"))
-            continue
-        for c, gb in enumerate(tires):
-            treads[(d, c)] = gb
 
     results = {}
-    _label_treads(treads, results)
+    _label_treads(treads, results, tread_meta=tread_meta)
     cap_cuts = _cap_cut(G, emb, source, levels, results)
     cut_graph, labels, warnings = _assemble_cut_graph(M, results, cap_cuts=cap_cuts)
     return {
         "n": n, "seed": seed, "graph_seed": graph_seed,
         "min_degree": min(dict(G.degree()).values()),
+        "connectivity": nx.node_connectivity(G),
         "G": G, "M": M, "source": source,
-        "treads": treads, "results": results, "cap_cuts": cap_cuts,
+        "treads": treads, "tread_meta": tread_meta,
+        "results": results, "cap_cuts": cap_cuts,
         "skipped": skipped,
         "cut_graph": cut_graph, "labels": labels, "warnings": warnings,
     }
@@ -370,6 +536,7 @@ def to_json(result: dict) -> dict:
     return {
         "n": result["n"], "seed": result["seed"],
         "graph_seed": result["graph_seed"], "min_degree": result["min_degree"],
+        "connectivity": result["connectivity"],
         "source": result["source"],
         "graph_edges": sorted([int(u), int(v)] for u, v in result["G"].edges()),
         "medial_vertices": result["M"].number_of_nodes(),
@@ -397,7 +564,8 @@ def summary(result: dict) -> str:
     lines = [
         f"random maximal planar graph: n={result['n']} requested seed={result['seed']} "
         f"graph seed={result['graph_seed']} "
-        f"({result['G'].number_of_edges()} edges, min degree {result['min_degree']})",
+        f"({result['G'].number_of_edges()} edges, "
+        f"connectivity {result['connectivity']}, min degree {result['min_degree']})",
         f"medial graph M(G): {result['M'].number_of_nodes()} vertices",
         f"level source: vertex {result['source']}",
         f"recognised tires (depth, component): {sorted(res)}",
@@ -427,16 +595,20 @@ def main() -> None:
     parser.add_argument("--seed", type=int, default=0)
     parser.add_argument("--flips", type=int, default=400,
                         help="number of random diagonal flips when building G")
-    parser.add_argument("--min-degree", type=int, default=5,
+    parser.add_argument("--min-connectivity", type=int, default=5,
-                        help="reject random graphs below this minimum degree")
+                        help="reject random graphs below this node connectivity")
+    parser.add_argument("--min-degree", type=int, default=None,
+                        help="compatibility alias; also raises min-connectivity")
     parser.add_argument("--attempts", type=int, default=1000,
-                        help="number of consecutive seeds to try for --min-degree")
+                        help="number of consecutive seeds to try for sampling")
     parser.add_argument("--json", metavar="PATH",
                         help="write the full result as JSON to PATH")
     args = parser.parse_args()
 
     result = run_experiment(n=args.n, seed=args.seed, flips=args.flips,
-                            min_degree=args.min_degree, attempts=args.attempts)
+                            min_connectivity=args.min_connectivity,
+                            min_degree=args.min_degree,
+                            attempts=args.attempts)
     print(summary(result))
     if args.json:
         with open(args.json, "w") as fh:

papers/medial_tire_cuts/lib/medial_tire_cut_labelling.py

@@ -205,7 +205,8 @@ def label_and_cut(graph: FullMedialTireGraph, entry_edge: int,
 
 
 def up_apex_cuts(graph: FullMedialTireGraph, entry_edge: int,
-                 bij: Mapping[str, object] | None = None) -> dict[int, object]:
+                 bij: Mapping[str, object] | None = None,
+                 shared_apexes: set[object] | None = None) -> dict[int, object]:
     """Up-tooth apex cuts prescribed after the walk-depth traversal.
 
     The returned dict maps each cut up-tooth edge to the apex vertex to
@@ -218,11 +219,12 @@ def up_apex_cuts(graph: FullMedialTireGraph, entry_edge: int,
         i: (bij[f"u{i}"] if bij is not None else graph.apex_of_edge(i))
         for i in graph.up_edges
     }
+    shared_apexes = shared_apexes or set()
     multiplicity = Counter(apex_by_edge.values())
     return {
         i: apex
         for i, apex in apex_by_edge.items()
-        if i != entry_edge and multiplicity[apex] == 1
+        if i != entry_edge and multiplicity[apex] == 1 and apex not in shared_apexes
     }
 
 

papers/medial_tire_decompositions_of_plane_triangulations/lib/draw_random_medial_tire_decompositions.py

@@ -812,7 +812,10 @@ def run(args: argparse.Namespace):
     out_dir.mkdir(parents=True, exist_ok=True)
 
     if args.graph6:
-        graphs = [nx.from_graph6_bytes(args.graph6.encode())]
+        graph = nx.from_graph6_bytes(args.graph6.encode())
+        if nx.node_connectivity(graph) < 5:
+            raise ValueError("--graph6 graph must be 5-connected")
+        graphs = [graph]
         if args.source is None:
             raise ValueError("--source is required with --graph6")
         sources = [args.source]