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