Augment same-level faces before medial tire extraction

papers/medial_tire_decompositions_of_plane_triangulations/experiments/branching_medial_tire_decomposition/random_c5_n30_medial_tire_decomposition_1.md

@@ -1,19 +1,25 @@
 # Random medial tire decomposition 1
 
-- vertices: 30
+- original vertices: 30
-- edges: 84
+- original edges: 84
-- node connectivity: 5
+- original node connectivity: 5
+- augmented vertices: 33
+- augmented edges: 93
+- same-level faces filled: 3
 - source vertex: 14
 - tire-tree nodes: 4
 - tire-tree edges: 3
 
-| node | depth | faces | annular | up | down | bites | full medial tires |
+| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |
 |--:|--:|--:|--:|--:|--:|--:|--:|
 | T0 | 2 | 23 | 23 | 10 | 13 | 0 | 1 |
 | T0.0 |  |  | | 10 | 13 | - | `UDDUDUUDDUDUDUDUDDDUUDD` |
 | T1 | 1 | 15 | 15 | 5 | 10 | 0 | 1 |
 | T1.0 |  |  | | 5 | 10 | - | `UDDDUDDUDUDDDUD` |
-| T2 | 3 | 8 | 5 | 11 | 0 | 0 | 1 |
+| T2 | 3 | 14 | 14 | 11 | 0 | 0 | 4 |
-| T2.0 |  |  | | 5 | 0 | - | `UUUUU` |
+| T2.0 |  |  | | 3 | 0 | - | `UUU` |
+| T2.1 |  |  | | 3 | 0 | - | `UUU` |
+| T2.2 |  |  | | 5 | 0 | - | `UUUUU` |
+| T2.3 |  |  | | 3 | 0 | - | `UUU` |
 | T3 | 3 | 5 | 5 | 5 | 0 | 0 | 1 |
 | T3.0 |  |  | | 5 | 0 | - | `UUUUU` |

papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_random_medial_tire_decompositions.py

@@ -52,6 +52,13 @@ class TreadNode:
     tires: tuple[tuple[FullMedialTireGraph, dict], ...]
 
 
+@dataclass(frozen=True)
+class Augmentation:
+    graph: nx.Graph
+    added_vertices: tuple[int, ...]
+    filled_faces: tuple[tuple[int, tuple[int, int, int], int], ...]
+
+
 def sample_plantri_graphs(n: int, count: int, seed: int, scan_limit: int) -> list[nx.Graph]:
     cmd = [str(REPO_ROOT / "plantri" / "plantri"), "-g", "-c5", str(n)]
     rng = random.Random(seed)
@@ -111,6 +118,39 @@ def edge_face_data(faces):
     return face_edges, edge_faces
 
 
+def augment_same_level_faces(g: nx.Graph, source: int) -> Augmentation:
+    """Stack a new vertex into every facial triangle with one BFS level.
+
+    If a triangular face has all three vertices at level d, the new vertex is
+    adjacent to those three vertices and therefore has level d + 1.  This turns
+    the same-level region into three adjacent-level tread faces before the tire
+    decomposition is extracted.
+    """
+    levels = nx.single_source_shortest_path_length(g, source)
+    faces = triangular_faces(g)
+    augmented = g.copy()
+    next_vertex = max(augmented.nodes()) + 1
+    added = []
+    filled = []
+
+    for face in faces:
+        face_levels = {levels[v] for v in face}
+        if len(face_levels) != 1:
+            continue
+        new_vertex = next_vertex
+        next_vertex += 1
+        augmented.add_node(new_vertex)
+        augmented.add_edges_from((new_vertex, v) for v in face)
+        added.append(new_vertex)
+        filled.append((new_vertex, tuple(face), next(iter(face_levels))))
+
+    return Augmentation(
+        graph=augmented,
+        added_vertices=tuple(added),
+        filled_faces=tuple(filled),
+    )
+
+
 def depth_components(faces, face_edges, edge_faces, levels):
     depths = [min(levels[v] for v in face) for face in faces]
     dual_adj: dict[int, set[int]] = defaultdict(set)
@@ -169,10 +209,12 @@ def tread_from_component(faces, levels, face_indices):
     }
 
 
-def build_tire_tree(g: nx.Graph, source: int):
+def build_tire_tree(g: nx.Graph, source: int, augment: bool = True):
-    faces = triangular_faces(g)
+    augmentation = augment_same_level_faces(g, source) if augment else Augmentation(g, (), ())
+    work_graph = augmentation.graph
+    faces = triangular_faces(work_graph)
     face_edges, edge_faces = edge_face_data(faces)
-    levels = nx.single_source_shortest_path_length(g, source)
+    levels = nx.single_source_shortest_path_length(work_graph, source)
     comps, depths, dual_adj = depth_components(faces, face_edges, edge_faces, levels)
     comp_of_face = {}
     for comp_idx, (_depth, face_indices) in enumerate(comps):
@@ -215,7 +257,7 @@ def build_tire_tree(g: nx.Graph, source: int):
                     parent_candidates.add(comp_to_node[other_comp])
         for parent in parent_candidates:
             tree_edges.add((parent, child))
-    return faces, levels, nodes, sorted(tree_edges)
+    return augmentation, faces, levels, nodes, sorted(tree_edges)
 
 
 def graph_layout(g: nx.Graph):
@@ -225,24 +267,37 @@ def graph_layout(g: nx.Graph):
         return nx.spring_layout(g, seed=0)
 
 
-def draw_base_graph(ax, g, levels, source):
+def draw_base_graph(ax, g, levels, source, added_vertices=()):
     pos = graph_layout(g)
     max_level = max(levels.values())
     cmap = plt.get_cmap("viridis", max_level + 1)
     node_colors = [cmap(levels[v]) for v in g.nodes()]
     nx.draw_networkx_edges(g, pos, ax=ax, edge_color="#cbd5e1", width=0.8)
+    added_set = set(added_vertices)
     nx.draw_networkx_nodes(
         g,
         pos,
         ax=ax,
         node_color=node_colors,
-        node_size=[150 if v == source else 72 for v in g.nodes()],
+        node_size=[
-        edgecolors=["#dc2626" if v == source else "#111827" for v in g.nodes()],
+            150 if v == source else 96 if v in added_set else 72
-        linewidths=[1.8 if v == source else 0.45 for v in g.nodes()],
+            for v in g.nodes()
+        ],
+        edgecolors=[
+            "#dc2626" if v == source else "#7c3aed" if v in added_set else "#111827"
+            for v in g.nodes()
+        ],
+        linewidths=[
+            1.8 if v == source else 1.2 if v in added_set else 0.45
+            for v in g.nodes()
+        ],
     )
     labels = {v: str(v) for v in g.nodes()}
     nx.draw_networkx_labels(g, pos, labels=labels, ax=ax, font_size=5)
-    ax.set_title(f"G, source {source}; vertex levels 0..{max_level}", fontsize=10)
+    ax.set_title(
+        f"Augmented G, source {source}; vertex levels 0..{max_level}",
+        fontsize=10,
+    )
     ax.set_aspect("equal")
     ax.axis("off")
 
@@ -374,52 +429,69 @@ def draw_medial_tire_grid(fig, outer_spec, nodes):
             ax.axis("off")
 
 
-def write_index(path: Path, graph_idx: int, source: int, g: nx.Graph, nodes, tree_edges):
+def write_index(
+    path: Path,
+    graph_idx: int,
+    source: int,
+    original_graph: nx.Graph,
+    augmentation: Augmentation,
+    nodes,
+    tree_edges,
+):
+    g = augmentation.graph
     lines = [
         f"# Random medial tire decomposition {graph_idx}",
         "",
-        f"- vertices: {g.number_of_nodes()}",
+        f"- original vertices: {original_graph.number_of_nodes()}",
-        f"- edges: {g.number_of_edges()}",
+        f"- original edges: {original_graph.number_of_edges()}",
-        f"- node connectivity: {nx.node_connectivity(g)}",
+        f"- original node connectivity: {nx.node_connectivity(original_graph)}",
+        f"- augmented vertices: {g.number_of_nodes()}",
+        f"- augmented edges: {g.number_of_edges()}",
+        f"- same-level faces filled: {len(augmentation.added_vertices)}",
         f"- source vertex: {source}",
         f"- tire-tree nodes: {len(nodes)}",
         f"- tire-tree edges: {len(tree_edges)}",
         "",
-        "| node | depth | faces | annular | up | down | bites | full medial tires |",
+        "| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |",
         "|--:|--:|--:|--:|--:|--:|--:|--:|",
     ]
     for node in nodes:
+        singleton_down = set(node.down) - set(node.bites)
         lines.append(
             f"| T{node.idx} | {node.depth} | {len(node.face_indices)} | "
-            f"{len(node.annular)} | {len(node.up)} | {len(node.down)} | "
+            f"{len(node.annular)} | {len(node.up)} | {len(singleton_down)} | "
             f"{len(node.bites)} | {len(node.tires)} |"
         )
         for comp_idx, (tire, _bij) in enumerate(node.tires):
             bites = ",".join(f"({i},{j})" for i, j in sorted(tire.bites)) or "-"
             lines.append(
                 f"| T{node.idx}.{comp_idx} |  |  | | {len(tire.up_edges)} | "
-                f"{len(tire.down_edges)} | {bites} | `{tire.tooth_word}` |"
+                f"{len(tire.singleton_down_edges)} | {bites} | `{tire.tooth_word}` |"
             )
     path.write_text("\n".join(lines) + "\n")
 
 
-def draw_case(out_dir: Path, graph_idx: int, g: nx.Graph, source: int):
+def draw_case(out_dir: Path, graph_idx: int, g: nx.Graph, source: int, augment: bool = True):
-    _faces, levels, nodes, tree_edges = build_tire_tree(g, source)
+    augmentation, _faces, levels, nodes, tree_edges = build_tire_tree(g, source, augment=augment)
+    work_graph = augmentation.graph
     fig = plt.figure(figsize=(17, 10))
     spec = fig.add_gridspec(2, 2, width_ratios=[1.15, 1.0], height_ratios=[1.0, 1.25])
     ax_graph = fig.add_subplot(spec[0, 0])
     ax_tree = fig.add_subplot(spec[1, 0])
-    draw_base_graph(ax_graph, g, levels, source)
+    draw_base_graph(ax_graph, work_graph, levels, source, augmentation.added_vertices)
     draw_tire_tree(ax_tree, nodes, tree_edges)
     draw_medial_tire_grid(fig, spec[:, 1], nodes)
     fig.suptitle(
         f"Random 5-connected maximal planar graph {graph_idx}: "
-        f"n={g.number_of_nodes()}, source={source}",
+        f"n={g.number_of_nodes()} (+{len(augmentation.added_vertices)}), "
+        f"source={source}",
         fontsize=13,
     )
     legend = [
         Line2D([0], [0], marker="o", color="w", label="source",
                markerfacecolor="#fde68a", markeredgecolor="#dc2626", markersize=8),
+        Line2D([0], [0], marker="o", color="w", label="inserted vertex",
+               markerfacecolor="#fde68a", markeredgecolor="#7c3aed", markersize=8),
         Line2D([0], [0], color="black", lw=1.3, label="annular cycle A(T)"),
         Line2D([0], [0], marker="o", color="w", label="up tooth",
                markerfacecolor="#2563eb", markersize=6),
@@ -441,6 +513,7 @@ def draw_case(out_dir: Path, graph_idx: int, g: nx.Graph, source: int):
         graph_idx,
         source,
         g,
+        augmentation,
         nodes,
         tree_edges,
     )
@@ -462,7 +535,9 @@ def run(args: argparse.Namespace):
         sources = [rng.choice(list(graph.nodes())) for graph in graphs]
 
     for i, (graph, source) in enumerate(zip(graphs, sources), start=1):
-        png, pdf, node_count, tire_count = draw_case(out_dir, i, graph, source)
+        png, pdf, node_count, tire_count = draw_case(
+            out_dir, i, graph, source, augment=not args.no_augment_same_level_faces
+        )
         print(
             f"case {i}: source={source}, connectivity={nx.node_connectivity(graph)}, "
             f"tire nodes={node_count}, full medial tires={tire_count}"
@@ -479,6 +554,11 @@ def main():
     parser.add_argument("--scan-limit", type=int, default=500)
     parser.add_argument("--graph6", help="draw this graph6 graph instead of sampling")
     parser.add_argument("--source", type=int, help="source vertex for --graph6")
+    parser.add_argument(
+        "--no-augment-same-level-faces",
+        action="store_true",
+        help="skip the same-level-face vertex insertion step",
+    )
     parser.add_argument(
         "--out-dir",
         default=str(PAPER_DIR / "experiments" / "random_medial_tire_decompositions"),

papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_1.md

@@ -1,18 +1,22 @@
 # Random medial tire decomposition 1
 
-- vertices: 30
+- original vertices: 30
-- edges: 84
+- original edges: 84
-- node connectivity: 5
+- original node connectivity: 5
+- augmented vertices: 31
+- augmented edges: 87
+- same-level faces filled: 1
 - source vertex: 9
 - tire-tree nodes: 3
 - tire-tree edges: 2
 
-| node | depth | faces | annular | up | down | bites | full medial tires |
+| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |
 |--:|--:|--:|--:|--:|--:|--:|--:|
 | T0 | 1 | 16 | 16 | 6 | 10 | 0 | 1 |
 | T0.0 |  |  | | 6 | 10 | - | `DUDUDUDDUDDDUDDU` |
 | T1 | 2 | 20 | 20 | 10 | 10 | 0 | 1 |
 | T1.0 |  |  | | 10 | 10 | - | `UUDUUDUDDUDUDUDUUDDD` |
-| T2 | 3 | 14 | 13 | 12 | 1 | 1 | 2 |
+| T2 | 3 | 16 | 16 | 12 | 0 | 1 | 3 |
-| T2.0 |  |  | | 6 | 2 | (1,5) | `UDUUUDUU` |
+| T2.0 |  |  | | 6 | 0 | (1,5) | `UDUUUDUU` |
-| T2.1 |  |  | | 5 | 0 | - | `UUUUU` |
+| T2.1 |  |  | | 3 | 0 | - | `UUU` |
+| T2.2 |  |  | | 5 | 0 | - | `UUUUU` |

papers/medial_tire_decompositions_of_plane_triangulations/experiments/random_medial_tire_decompositions/random_c5_n30_medial_tire_decomposition_2.md

@@ -1,17 +1,23 @@
 # Random medial tire decomposition 2
 
-- vertices: 30
+- original vertices: 30
-- edges: 84
+- original edges: 84
-- node connectivity: 5
+- original node connectivity: 5
+- augmented vertices: 32
+- augmented edges: 90
+- same-level faces filled: 2
 - source vertex: 4
-- tire-tree nodes: 3
+- tire-tree nodes: 4
-- tire-tree edges: 2
+- tire-tree edges: 3
 
-| node | depth | faces | annular | up | down | bites | full medial tires |
+| node | depth | faces | annular | up | singleton down | bite apexes | full medial tires |
 |--:|--:|--:|--:|--:|--:|--:|--:|
 | T0 | 1 | 16 | 16 | 7 | 9 | 0 | 1 |
 | T0.0 |  |  | | 7 | 9 | - | `DUDUDUDUDUDDUDUD` |
 | T1 | 2 | 17 | 17 | 9 | 8 | 0 | 1 |
 | T1.0 |  |  | | 9 | 8 | - | `UUUDDUDUDUDUDDUUD` |
-| T2 | 3 | 14 | 14 | 8 | 5 | 1 | 1 |
+| T2 | 3 | 14 | 14 | 8 | 4 | 1 | 1 |
-| T2.0 |  |  | | 8 | 6 | (1,5) | `DDUUUDDUUDUDUU` |
+| T2.0 |  |  | | 8 | 4 | (1,5) | `DDUUUDDUUDUDUU` |
+| T3 | 4 | 6 | 6 | 5 | 0 | 0 | 2 |
+| T3.0 |  |  | | 3 | 0 | - | `UUU` |
+| T3.1 |  |  | | 3 | 0 | - | `UUU` |