Replace plane_depth_labelling with plane_depth_sequencing paper and script, remove unused lib modules

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

plane_depth_sequencing.py

@@ -0,0 +1,65 @@
+"""Plane depth sequencing on maximal planar graphs."""
+import random
+from typing import Any, TypedDict
+from sage.all import Graph, graphs  # type: ignore[attr-defined]  # pylint: disable=no-name-in-module
+
+
+class DeeplyEmbeddedGraph(TypedDict):
+    graph: Graph
+    outer_cycle: list[Any]
+    plane_depth_labelling: dict[Any, int]
+    deep_embedding: Graph
+
+
+def get_plane_depth_labelling(g: Graph, outer_cycle: list[Any]) -> dict[Any, int]:
+    """Return the plane depth of each vertex relative to the given outer cycle."""
+    # equivalent to the commented out naive implementation:
+    # return {v: min(g.distance(v, u) for u in outer_cycle) for v in g.vertices()}
+    source = max(g.vertices()) + 1
+    g_prime = g.copy()
+    g_prime.add_vertex(source)
+    g_prime.add_edges([(source, v) for v in outer_cycle])
+    distances = g_prime.breadth_first_search(source, report_distance=True)
+    return {v: d - 1 for v, d in distances if v != source}
+
+
+def deep_embedding(g: Graph, outer_cycle: list[Any], plane_depth_labelling: dict[Any, int] | None = None) -> Graph:
+    """
+    Return the deep embedding of g relative to outer_cycle.
+
+    For every interior triangular face whose three vertices all have the same
+    plane depth, a new vertex is added adjacent to each vertex of that face.
+    """
+    if plane_depth_labelling is None:
+        plane_depth_labelling = get_plane_depth_labelling(g, outer_cycle)
+    outer_vertices = set(outer_cycle)
+
+    embedding = g.get_embedding()
+    if embedding is None:
+        g.is_planar(set_embedding=True)
+        embedding = g.get_embedding()
+
+    g_prime = g.copy()
+    new_vertex = max(g.vertices()) + 1
+
+    for face in g.faces(embedding):
+        face_vertices = [u for u, v in face]
+        if set(face_vertices) == outer_vertices:
+            continue
+        if plane_depth_labelling[face_vertices[0]] == plane_depth_labelling[face_vertices[1]] == plane_depth_labelling[face_vertices[2]]:
+            g_prime.add_vertex(new_vertex)
+            g_prime.add_edges([(new_vertex, v) for v in face_vertices])
+            new_vertex += 1
+
+    return g_prime
+
+
+def generate_example(n: int) -> DeeplyEmbeddedGraph:
+    """Generate a random maximal planar graph of size n and return the triangulation, outer cycle, and deep embedding."""
+    g = graphs.RandomTriangulation(n)
+    g.is_planar(set_embedding=True)
+    embedding = g.get_embedding()
+    faces = g.faces(embedding)
+    outer_cycle = [u for u, v in random.choice(faces)]
+    plane_depth_labelling = get_plane_depth_labelling(g, outer_cycle)
+    return DeeplyEmbeddedGraph(graph=g, outer_cycle=outer_cycle, plane_depth_labelling=plane_depth_labelling, deep_embedding=deep_embedding(g, outer_cycle, plane_depth_labelling))