Add medial tire decomposition paper
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"""Compare full and reduced medial tire graphs on generated tires.
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The new medial decomposition paper defines:
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* full medial tire graph: the subgraph of M(G) induced by medial
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vertices corresponding to edges incident to tread triangles;
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* reduced medial tire graph: delete same-boundary medial edges and
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chord-only medial edges.
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For a tire tread inside an ambient triangulation, the medial edges
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visible in the tread come from annular triangular faces. This script
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checks whether any same-boundary medial edges are actually present in
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that model. It also compares against the older standalone drawing
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model, which added artificial outer/inner boundary faces.
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"""
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from __future__ import annotations
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import argparse
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import itertools
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import random
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from collections import Counter
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Edge = tuple[int, int]
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MedialEdge = tuple[Edge, Edge]
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def random_tire(m: int, k: int, n_chords: int = 0, seed: int | None = None) -> dict:
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"""Generate the same labelled annular tires used in earlier experiments."""
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rng = random.Random(seed)
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outer = list(range(m))
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inner = list(range(m, m + k))
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edges: set[Edge] = set()
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for i in range(m):
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edges.add(edge_key(outer[i], outer[(i + 1) % m]))
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for j in range(k):
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edges.add(edge_key(inner[j], inner[(j + 1) % k]))
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inner_chords = set()
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candidates = []
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for a in range(k):
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for b in range(a + 2, k):
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if not (a == 0 and b == k - 1):
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candidates.append((a, b))
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rng.shuffle(candidates)
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for a, b in candidates:
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if len(inner_chords) >= n_chords:
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break
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if any((a < a2 < b < b2) or (a2 < a < b2 < b) for a2, b2 in inner_chords):
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continue
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inner_chords.add((a, b))
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edges.add(edge_key(inner[a], inner[b]))
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edges.add(edge_key(outer[0], inner[0]))
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moves = ["O"] * m + ["I"] * k
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rng.shuffle(moves)
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triangles = []
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i, j = 0, 0
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for move in moves:
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if move == "O":
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tri = (outer[i % m], inner[j % k], outer[(i + 1) % m])
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triangles.append(tri)
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edges.add(edge_key(inner[j % k], outer[(i + 1) % m]))
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i += 1
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else:
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tri = (outer[i % m], inner[j % k], inner[(j + 1) % k])
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triangles.append(tri)
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edges.add(edge_key(outer[i % m], inner[(j + 1) % k]))
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j += 1
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return {
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"m": m,
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"k": k,
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"n_chords": len(inner_chords),
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"outer": outer,
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"inner": inner,
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"edges": sorted(edges),
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"triangles": triangles,
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"inner_chords": sorted(inner_chords),
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"lattice_path": "".join(moves),
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"seed": seed,
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}
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def tire_from_path(m: int, k: int, chords: tuple[tuple[int, int], ...], path: str) -> dict:
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outer = list(range(m))
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inner = list(range(m, m + k))
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edges: set[Edge] = set()
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for i in range(m):
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edges.add(edge_key(outer[i], outer[(i + 1) % m]))
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for j in range(k):
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edges.add(edge_key(inner[j], inner[(j + 1) % k]))
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for a, b in chords:
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edges.add(edge_key(inner[a], inner[b]))
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edges.add(edge_key(outer[0], inner[0]))
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triangles = []
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i, j = 0, 0
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for move in path:
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if move == "O":
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tri = (outer[i % m], inner[j % k], outer[(i + 1) % m])
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triangles.append(tri)
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edges.add(edge_key(inner[j % k], outer[(i + 1) % m]))
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i += 1
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else:
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tri = (outer[i % m], inner[j % k], inner[(j + 1) % k])
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triangles.append(tri)
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edges.add(edge_key(outer[i % m], inner[(j + 1) % k]))
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j += 1
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return {
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"m": m,
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"k": k,
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"n_chords": len(chords),
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"outer": outer,
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"inner": inner,
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"edges": sorted(edges),
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"triangles": triangles,
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"inner_chords": sorted(chords),
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"lattice_path": path,
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"seed": None,
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}
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def chord_crosses(c1: tuple[int, int], c2: tuple[int, int]) -> bool:
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a, b = c1
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c, d = c2
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return (a < c < b < d) or (c < a < d < b)
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def chord_sets(k: int, max_chords: int) -> list[tuple[tuple[int, int], ...]]:
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candidates = []
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for a in range(k):
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for b in range(a + 2, k):
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if not (a == 0 and b == k - 1):
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candidates.append((a, b))
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out = [()]
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def rec(start: int, chosen: tuple[tuple[int, int], ...]) -> None:
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if len(chosen) >= max_chords:
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return
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for idx in range(start, len(candidates)):
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chord = candidates[idx]
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if any(chord_crosses(chord, old) for old in chosen):
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continue
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nxt = chosen + (chord,)
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out.append(nxt)
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rec(idx + 1, nxt)
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rec(0, ())
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return out
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def lattice_paths(m: int, k: int):
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for o_positions in itertools.combinations(range(m + k), m):
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o_set = set(o_positions)
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yield "".join("O" if idx in o_set else "I" for idx in range(m + k))
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def edge_key(u: int, v: int) -> Edge:
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return tuple(sorted((u, v)))
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def face_edges(face: tuple[int, ...]) -> list[Edge]:
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return [edge_key(face[i], face[(i + 1) % len(face)]) for i in range(len(face))]
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def is_cycle_edge(edge: Edge, cycle: list[int]) -> bool:
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cycle_set = set(cycle)
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if not set(edge) <= cycle_set:
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return False
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n = len(cycle)
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idx = {v: i for i, v in enumerate(cycle)}
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a, b = idx[edge[0]], idx[edge[1]]
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return (a - b) % n in (1, n - 1)
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def is_inner_chord(edge: Edge, m: int, k: int) -> bool:
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u, v = edge
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if not (m <= u < m + k and m <= v < m + k):
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return False
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a, b = u - m, v - m
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d = abs(a - b)
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return min(d, k - d) != 1
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def suppress_reason(e1: Edge, e2: Edge, tire: dict) -> str | None:
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outer = tire["outer"]
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inner = tire["inner"]
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if is_cycle_edge(e1, outer) and is_cycle_edge(e2, outer):
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return "outer_boundary"
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if is_cycle_edge(e1, inner) and is_cycle_edge(e2, inner):
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return "inner_boundary"
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m, k = tire["m"], tire["k"]
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if is_inner_chord(e1, m, k) or is_inner_chord(e2, m, k):
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return "inner_chord"
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return None
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def medial_from_faces(faces: list[tuple[int, ...]], retained: set[Edge]) -> set[MedialEdge]:
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medial_edges: set[MedialEdge] = set()
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for face in faces:
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boundary = [e for e in face_edges(face) if e in retained]
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if len(boundary) < 2:
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continue
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for i, e in enumerate(boundary):
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nxt = boundary[(i + 1) % len(boundary)]
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if e != nxt:
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medial_edges.add(tuple(sorted((e, nxt))))
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return medial_edges
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def compare_tire(tire: dict, *, standalone_boundary_faces: bool) -> dict:
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annular_faces = [tuple(tri) for tri in tire["triangles"]]
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faces = list(annular_faces)
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if standalone_boundary_faces:
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faces.append(tuple(tire["outer"]))
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faces.append(tuple(reversed(tire["inner"])))
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# Definition 3.1 includes edges incident to at least one tread triangle.
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retained = {e for face in annular_faces for e in face_edges(face)}
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full_edges = medial_from_faces(faces, retained)
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removed = {me for me in full_edges if suppress_reason(me[0], me[1], tire)}
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reduced_edges = full_edges - removed
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reasons = Counter(suppress_reason(me[0], me[1], tire) for me in removed)
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reasons.pop(None, None)
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return {
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"vertices": len(retained),
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"full_edges": len(full_edges),
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"reduced_edges": len(reduced_edges),
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"removed": len(removed),
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"reasons": reasons,
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"examples": sorted(removed)[:5],
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}
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def run_sweep(args: argparse.Namespace) -> None:
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ambient_cases = 0
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ambient_differ = []
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standalone_cases = 0
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standalone_differ = []
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ambient_reasons: Counter[str] = Counter()
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standalone_reasons: Counter[str] = Counter()
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max_chords = args.max_chords
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for m in range(args.min_cycle, args.max_cycle + 1):
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for k in range(args.min_cycle, args.max_cycle + 1):
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for chords in range(max_chords + 1):
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for seed in range(args.seeds):
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tire = random_tire(m=m, k=k, n_chords=chords, seed=seed)
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ambient = compare_tire(tire, standalone_boundary_faces=False)
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ambient_cases += 1
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ambient_reasons.update(ambient["reasons"])
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if ambient["removed"]:
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ambient_differ.append((m, k, chords, seed, tire, ambient))
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standalone = compare_tire(tire, standalone_boundary_faces=True)
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standalone_cases += 1
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standalone_reasons.update(standalone["reasons"])
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if standalone["removed"]:
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standalone_differ.append((m, k, chords, seed, tire, standalone))
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print("ambient tread-face model")
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print(f" cases checked: {ambient_cases}")
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print(f" cases where full != reduced: {len(ambient_differ)}")
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print(f" removed-edge reasons: {dict(sorted(ambient_reasons.items()))}")
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if ambient_differ:
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m, k, chords, seed, tire, result = ambient_differ[0]
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print(" first difference:")
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print(f" m={m} k={k} requested_chords={chords} seed={seed}")
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print(f" path={tire['lattice_path']} chords={tire['inner_chords']}")
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print(f" removed examples={result['examples']}")
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print()
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print("standalone tire-with-boundary-faces model")
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print(f" cases checked: {standalone_cases}")
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print(f" cases where full != reduced: {len(standalone_differ)}")
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print(f" removed-edge reasons: {dict(sorted(standalone_reasons.items()))}")
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if standalone_differ:
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m, k, chords, seed, tire, result = standalone_differ[0]
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print(" first difference:")
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print(f" m={m} k={k} requested_chords={chords} seed={seed}")
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print(f" path={tire['lattice_path']} chords={tire['inner_chords']}")
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print(f" full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}")
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print(f" removed examples={result['examples']}")
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def run_exhaustive(args: argparse.Namespace) -> None:
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ambient_cases = 0
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ambient_differ = []
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standalone_cases = 0
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standalone_differ = []
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for m in range(args.min_cycle, args.max_cycle + 1):
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for k in range(args.min_cycle, args.max_cycle + 1):
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for chords in chord_sets(k, args.max_chords):
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for path in lattice_paths(m, k):
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tire = tire_from_path(m, k, chords, path)
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ambient = compare_tire(tire, standalone_boundary_faces=False)
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ambient_cases += 1
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if ambient["removed"]:
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ambient_differ.append((m, k, chords, path, ambient))
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standalone = compare_tire(tire, standalone_boundary_faces=True)
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standalone_cases += 1
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if standalone["removed"]:
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standalone_differ.append((m, k, chords, path, standalone))
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print("exhaustive ambient tread-face model")
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print(f" cases checked: {ambient_cases}")
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print(f" cases where full != reduced: {len(ambient_differ)}")
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if ambient_differ:
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m, k, chords, path, result = ambient_differ[0]
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print(" first difference:")
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print(f" m={m} k={k} chords={chords} path={path}")
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print(f" removed examples={result['examples']}")
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print()
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print("exhaustive standalone tire-with-boundary-faces model")
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print(f" cases checked: {standalone_cases}")
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print(f" cases where full != reduced: {len(standalone_differ)}")
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if standalone_differ:
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m, k, chords, path, result = standalone_differ[0]
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print(" first difference:")
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print(f" m={m} k={k} chords={chords} path={path}")
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print(f" full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}")
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print(f" removed examples={result['examples']}")
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def main() -> None:
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parser = argparse.ArgumentParser()
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parser.add_argument("--min-cycle", type=int, default=3)
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parser.add_argument("--max-cycle", type=int, default=8)
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parser.add_argument("--max-chords", type=int, default=3)
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parser.add_argument("--seeds", type=int, default=50)
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parser.add_argument("--exhaustive", action="store_true")
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args = parser.parse_args()
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if args.exhaustive:
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run_exhaustive(args)
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else:
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run_sweep(args)
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if __name__ == "__main__":
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main()
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+82
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# Full vs Reduced Medial Tire Findings
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Question: do Definition 3.1 (full medial tire graph) and Definition 3.2
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(reduced medial tire graph) differ?
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## Experiment
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Script:
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```bash
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python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
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```
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The script compares two models.
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- Ambient tread-face model: medial edges are contributed by annular
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triangular faces of the tire tread inside the ambient triangulation.
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- Standalone tire-with-boundary-faces model: the outer and inner
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boundary walks are also treated as faces, as in the older drawing
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script.
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## Random Sweep
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Command:
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```bash
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python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
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```
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Result:
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```text
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ambient tread-face model
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cases checked: 7200
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cases where full != reduced: 0
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removed-edge reasons: {}
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standalone tire-with-boundary-faces model
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cases checked: 7200
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cases where full != reduced: 7200
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removed-edge reasons: {'inner_boundary': 39600, 'outer_boundary': 39600}
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first difference:
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m=3 k=3 requested_chords=0 seed=0
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path=IOOOII chords=[]
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full_edges=24 reduced_edges=18
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removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
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```
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## Exhaustive Small Sweep
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Command:
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```bash
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python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py --exhaustive --max-cycle 5 --max-chords 2
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```
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Result:
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```text
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exhaustive ambient tread-face model
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cases checked: 5578
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cases where full != reduced: 0
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exhaustive standalone tire-with-boundary-faces model
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cases checked: 5578
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cases where full != reduced: 5578
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first difference:
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m=3 k=3 chords=() path=OOOIII
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full_edges=24 reduced_edges=18
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removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
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```
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## Interpretation
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For the intended ambient-triangulation definition, the experiments
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support the suspicion that Definition 3.1 and Definition 3.2 coincide:
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same-boundary medial edges do not arise from annular triangular tread
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faces, and inner chords are not incident to tread triangles.
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They differ only in the standalone tire-with-boundary-faces model,
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where the artificial outer and inner boundary faces create medial edges
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between consecutive boundary edges.
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@@ -0,0 +1,34 @@
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\relax
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\citation{bauerfeld-nested-tire-decompositions}
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\citation{bauerfeld-nested-tire-decompositions}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background}}{1}{}\protected@file@percent }
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\citation{bauerfeld-nested-tire-decompositions}
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\citation{bauerfeld-nested-tire-decompositions}
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\newlabel{def:medial-graph}{{2.1}{2}}
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\newlabel{prop:medial-dual-invariance}{{2.3}{2}}
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\newlabel{cor:tait-medial}{{2.4}{2}}
|
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||||
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|
||||
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|
||||
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||||
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||||
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||||
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%% filename: amsart-template.tex
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||||
%% American Mathematical Society
|
||||
%% AMS-LaTeX v.2 template for use with amsart
|
||||
%% ====================================================================
|
||||
|
||||
\documentclass{amsart}
|
||||
|
||||
\usepackage{amssymb}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{backgrounds}
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||||
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
\newtheorem{corollary}[theorem]{Corollary}
|
||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
\newtheorem{conjecture}[theorem]{Conjecture}
|
||||
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{definition}[theorem]{Definition}
|
||||
\newtheorem{example}[theorem]{Example}
|
||||
\newtheorem{xca}[theorem]{Exercise}
|
||||
|
||||
\theoremstyle{remark}
|
||||
\newtheorem{remark}[theorem]{Remark}
|
||||
|
||||
\numberwithin{equation}{section}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\title{Medial Tire Decompositions of Plane Triangulations}
|
||||
|
||||
% author one information
|
||||
\author{Eric Bauerfeld}
|
||||
\address{}
|
||||
\curraddr{}
|
||||
\email{}
|
||||
\thanks{}
|
||||
|
||||
\subjclass[2010]{Primary }
|
||||
|
||||
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}
|
||||
|
||||
\date{}
|
||||
|
||||
\dedicatory{}
|
||||
|
||||
\begin{abstract}
|
||||
We use the nested tire decomposition of a plane triangulation to induce
|
||||
a decomposition of its full medial graph into medial tire subgraphs.
|
||||
For a plane triangulation $G$, the medial graph $M(G)$ is naturally
|
||||
isomorphic to the medial graph of the planar dual $G^*$, and proper
|
||||
$3$-vertex-colourings of $M(G)$ are equivalent to proper
|
||||
$3$-edge-colourings of the cubic dual. Thus Tait's reformulation of
|
||||
the Four Colour Theorem may be studied through proper vertex
|
||||
$3$-colourings of medial subgraphs. We define medial tire pieces,
|
||||
their boundary-state restriction relations, and a chain-pigeonhole
|
||||
conjecture for compatible medial boundary states across the tire tree.
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Introduction}
|
||||
|
||||
A classical theorem of Tait recasts the Four Colour Theorem in dual,
|
||||
edge-colouring terms: a plane triangulation $G$ is properly
|
||||
$4$-vertex-colourable if and only if its dual cubic graph $G^*$ is
|
||||
properly $3$-edge-colourable. The present paper records a medial
|
||||
version of this viewpoint. The vertices of the medial graph $M(G)$
|
||||
correspond to edges of $G$, and adjacency in $M(G)$ records
|
||||
consecutiveness of edges around vertices and faces of $G$. Since
|
||||
planar duality interchanges vertices and faces while preserving the
|
||||
edge set, $M(G)$ is naturally isomorphic to $M(G^*)$.
|
||||
|
||||
Consequently a proper vertex $3$-colouring of $M(G)$ is the same
|
||||
object as a proper edge $3$-colouring of $G^*$. This suggests another
|
||||
route toward the Four Colour Theorem: rather than colouring the dual
|
||||
cubic graph directly, decompose the full medial graph into local
|
||||
annular pieces and try to prove that their proper vertex
|
||||
$3$-colouring boundary restrictions always compose.
|
||||
|
||||
The structural input is the nested tire decomposition of
|
||||
\cite{bauerfeld-nested-tire-decompositions}. A level source in a plane
|
||||
triangulation determines a rooted tree of tire treads. Each tread is
|
||||
an annular triangulated region with an outer boundary, an inner
|
||||
outerplanar graph, and annular triangular faces. We show that this
|
||||
decomposition induces a decomposition of $M(G)$ into medial tire
|
||||
subgraphs. The boundary data of a medial tire are proper
|
||||
$3$-colourings of the medial vertices corresponding to boundary edges
|
||||
in the associated dual tire graph.
|
||||
|
||||
\section{Background}
|
||||
|
||||
Throughout, $G$ is a simple plane maximal planar graph with fixed
|
||||
embedding, and $G^*$ denotes its full planar dual. We use the level
|
||||
source, dual depth, tire graph, tire tread, and tire-tree terminology
|
||||
of~\cite{bauerfeld-nested-tire-decompositions}. In particular, a level
|
||||
source $S$ determines a rooted tire tree $\mathcal{T}(G,S)$ whose
|
||||
vertices are tire treads and whose parent-child relation records
|
||||
nested containment across level-cycle interfaces.
|
||||
|
||||
\begin{definition}[Medial graph]
|
||||
\label{def:medial-graph}
|
||||
Let $H$ be a plane graph. The \emph{medial graph} $M(H)$ has one
|
||||
vertex $m_e$ for each edge $e \in E(H)$. Two medial vertices
|
||||
$m_e,m_f$ are adjacent whenever $e$ and $f$ are consecutive in the
|
||||
cyclic order of edges around a vertex of $H$ or around a face of $H$.
|
||||
The embedding is the standard one obtained by placing $m_e$ at the
|
||||
midpoint of $e$ and drawing medial edges through the vertex- and
|
||||
face-corners of $H$.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
If $H$ has bridges or vertices of degree $1$, the usual medial
|
||||
construction may create parallel edges or loops depending on the
|
||||
chosen convention. In this paper the main application is to plane
|
||||
triangulations and their cubic planar duals, where the medial graph is
|
||||
a loopless $4$-regular plane graph.
|
||||
\end{remark}
|
||||
|
||||
\begin{proposition}[Medial dual invariance]
|
||||
\label{prop:medial-dual-invariance}
|
||||
Let $H$ be a connected plane graph and let $H^*$ be its planar dual.
|
||||
Then there is a natural plane-graph isomorphism
|
||||
\[
|
||||
M(H) \cong M(H^*).
|
||||
\]
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
Each edge $e \in E(H)$ corresponds to a unique dual edge $e^* \in
|
||||
E(H^*)$, giving a bijection $m_e \mapsto m_{e^*}$ between the vertices
|
||||
of $M(H)$ and $M(H^*)$. In $M(H)$ two vertices $m_e,m_f$ are adjacent
|
||||
exactly when $e$ and $f$ are consecutive around either a vertex or a
|
||||
face of $H$. Under duality, vertices and faces are interchanged, and
|
||||
the cyclic order of the corresponding dual edges around the dual face
|
||||
or dual vertex is the same up to reversal. Thus the same pairs are
|
||||
medial-adjacent in $M(H^*)$, and the midpoint construction identifies
|
||||
the two embedded medial graphs.
|
||||
\end{proof}
|
||||
|
||||
\begin{corollary}[Tait colourings as medial vertex colourings]
|
||||
\label{cor:tait-medial}
|
||||
Let $G$ be a simple plane triangulation. Proper vertex
|
||||
$3$-colourings of $M(G)$ are in natural bijection with proper
|
||||
$3$-edge-colourings of the cubic planar dual $G^*$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{proof}
|
||||
By Proposition~\ref{prop:medial-dual-invariance}, $M(G) \cong
|
||||
M(G^*)$. Vertices of $M(G^*)$ correspond to edges of $G^*$, and two
|
||||
such vertices are adjacent exactly when the corresponding dual edges
|
||||
are incident and consecutive around a vertex or face of $G^*$. Since
|
||||
$G^*$ is cubic, proper vertex $3$-colouring of $M(G^*)$ is therefore
|
||||
equivalent to assigning three colours to the edges of $G^*$ so that the
|
||||
three edges incident to each dual vertex receive pairwise distinct
|
||||
colours.
|
||||
\end{proof}
|
||||
|
||||
\section{Medial tire pieces}
|
||||
|
||||
\begin{definition}[Full medial tire graph]
|
||||
\label{def:full-medial-tire}
|
||||
Let $T$ be a tire tread in the tire tree $\mathcal{T}(G,S)$ supplied
|
||||
by~\cite{bauerfeld-nested-tire-decompositions}. The \emph{full medial
|
||||
tire graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of
|
||||
$M(G)$ induced by the medial vertices $m_e$ with $e$ an edge of $G$
|
||||
incident to at least one triangular face in the tread $T$. The medial
|
||||
vertices corresponding to annular edges of $T$ are called
|
||||
\emph{annular medial vertices}.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
In the ambient-triangulation setting, the full medial tire graph
|
||||
$\mathsf{M}(T)$ coincides with the omitted-edge medial tire graph
|
||||
studied in~\cite{bauerfeld-nested-tire-decompositions}. Indeed, the
|
||||
medial edges of $\mathsf{M}(T)$ are contributed by corners of annular
|
||||
triangular tread faces. Such a face contains at most one outer-boundary
|
||||
edge and at most one inner-boundary edge, so it does not contribute a
|
||||
medial edge between two outer-boundary edges or between two
|
||||
inner-boundary edges. Similarly, chords of the inner outerplanar graph
|
||||
lie outside the annular tread and are not incident to annular tread
|
||||
faces. Thus the deletion rule used for the earlier reduced medial tire
|
||||
graph removes no edges from the ambient object $\mathsf{M}(T)$.
|
||||
|
||||
The distinction only appears in the standalone drawing convention where
|
||||
the outer and inner boundary walks are added as artificial faces before
|
||||
forming a medial graph. Those artificial faces create same-boundary
|
||||
medial edges, and the reduced construction deletes them.
|
||||
\end{remark}
|
||||
|
||||
\begin{theorem}[Annular medial colour bound]
|
||||
\label{thm:annular-medial-colour-bound}
|
||||
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire tread with
|
||||
non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
|
||||
Let $A(T)$ be the subgraph of $\mathsf{M}(T)$ induced by the annular
|
||||
medial vertices. For a graph $H$, write $\operatorname{Col}_3(H)$ for
|
||||
the set of proper $3$-vertex-colourings of $H$. Then $A(T)$ is a cycle
|
||||
and
|
||||
\[
|
||||
|\operatorname{Col}_3(\mathsf{M}(T))|
|
||||
\;\leq\; |\operatorname{Col}_3(A(T))|.
|
||||
\]
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
Since the tread is a triangulated annulus with no vertices in its
|
||||
interior, each annular face has exactly one boundary edge, lying either
|
||||
on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular
|
||||
edges. As the annular faces are traversed cyclically around the tread,
|
||||
consecutive faces share one annular edge. Equivalently, the annular
|
||||
edges occur in a cyclic order in which each annular face contains two
|
||||
consecutive annular edges. Hence the subgraph of $\mathsf{M}(T)$
|
||||
induced by the annular medial vertices is a cycle.
|
||||
|
||||
Consider the restriction map from proper $3$-colourings of
|
||||
$\mathsf{M}(T)$ to colourings of this annular medial cycle $A(T)$. We
|
||||
claim that this map is injective. Let $x$ be a non-annular medial
|
||||
vertex. Then $x$ corresponds to an edge of $B_{\mathrm{out}}$ or
|
||||
$B_{\mathrm{in}}$: chords of $O$ are not incident to annular tread
|
||||
faces, and hence do not contribute vertices of $\mathsf{M}(T)$. This
|
||||
boundary edge is incident to a unique annular face of the tread, and
|
||||
the other two edges of that face are annular edges. Therefore $x$ is
|
||||
adjacent in $\mathsf{M}(T)$ to the two annular medial vertices
|
||||
corresponding to those two annular edges.
|
||||
|
||||
Those two annular medial vertices are adjacent to each other, because
|
||||
their annular edges are consecutive on the same triangular annular
|
||||
face. In any proper $3$-colouring they therefore receive two distinct
|
||||
colours, and $x$ is forced to receive the remaining third colour. Thus
|
||||
every non-annular medial vertex has its colour uniquely determined by
|
||||
the colouring of $A(T)$. Two colourings of $\mathsf{M}(T)$ with the
|
||||
same restriction to $A(T)$ are identical, so the restriction map is
|
||||
injective. The stated inequality follows.
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}[Boundary medial vertices]
|
||||
\label{def:boundary-medial-vertices}
|
||||
Let $T$ be a tire tread and let $\Gamma_T$ be the corresponding dual
|
||||
tire subgraph in $G^*$. A vertex $m_e \in V(\mathsf{M}(T))$ is an
|
||||
\emph{outer boundary medial vertex} if the corresponding dual edge
|
||||
$e^* \in E(G^*)$ lies on the outer boundary of $\Gamma_T$. It is an
|
||||
\emph{inner boundary medial vertex} if $e^*$ lies on the inner boundary
|
||||
of $\Gamma_T$. We write
|
||||
\[
|
||||
\partial_{\mathrm{out}}\mathsf{M}(T)
|
||||
\quad\text{and}\quad
|
||||
\partial_{\mathrm{in}}\mathsf{M}(T)
|
||||
\]
|
||||
for the two boundary sets.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Medial tire restriction relation]
|
||||
\label{def:medial-restriction-relation}
|
||||
Let $\mathrm{Col}_3(X)$ denote the set of proper vertex
|
||||
$3$-colourings of the induced subgraph on a vertex set $X$. The
|
||||
\emph{medial tire restriction relation} of $T$ is
|
||||
\[
|
||||
R_T \subseteq
|
||||
\mathrm{Col}_3(\partial_{\mathrm{out}}\mathsf{M}(T))
|
||||
\times
|
||||
\mathrm{Col}_3(\partial_{\mathrm{in}}\mathsf{M}(T)),
|
||||
\]
|
||||
where $(\alpha,\beta) \in R_T$ exactly when $\alpha \cup \beta$
|
||||
extends to a proper vertex $3$-colouring of $\mathsf{M}(T)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
The definition deliberately records boundary colourings on medial
|
||||
vertices corresponding to boundary edges in the dual tire graph. Under
|
||||
Corollary~\ref{cor:tait-medial}, these are precisely edge-colouring
|
||||
states on the boundary edges through which a dual tire piece meets its
|
||||
parent and children.
|
||||
\end{remark}
|
||||
|
||||
\section{Decomposition}
|
||||
|
||||
\begin{corollary}[Medial tire decomposition]
|
||||
\label{cor:medial-tire-decomposition}
|
||||
Let $G$ be a plane triangulation with level source $S$. The tire-tree
|
||||
decomposition $\mathcal{T}(G,S)$ of
|
||||
\cite{bauerfeld-nested-tire-decompositions} induces a rooted
|
||||
decomposition of the full medial graph $M(G)$ into full medial tire
|
||||
graphs $\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along
|
||||
their boundary medial vertex sets.
|
||||
\end{corollary}
|
||||
|
||||
\begin{proof}
|
||||
By the tire-tread partition theorem of
|
||||
\cite{bauerfeld-nested-tire-decompositions}, the bounded triangular
|
||||
faces of $G$ are partitioned into nested tire treads, with intersections
|
||||
between parent and child treads occurring only along their level-cycle
|
||||
interface data. Every edge of $G$ that is incident to a bounded face
|
||||
therefore belongs to the closure of at least one tire tread, and an
|
||||
edge lying in two closures lies on the interface between adjacent
|
||||
treads in the tire tree. Passing to $M(G)$ sends edges of $G$ to
|
||||
medial vertices. Thus each tread determines the induced subgraph
|
||||
$\mathsf{M}(T)$ on its incident edge set, and overlaps between two such
|
||||
subgraphs are exactly the medial vertices corresponding to interface
|
||||
edges, namely the appropriate boundary medial vertex sets.
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}[Compatible family of medial tire colourings]
|
||||
\label{def:compatible-family}
|
||||
A \emph{compatible family of medial tire colourings} on
|
||||
$\mathcal{T}(G,S)$ is a choice, for each tread $T$, of a proper
|
||||
vertex $3$-colouring $\varphi_T$ of $\mathsf{M}(T)$ such that whenever
|
||||
$T'$ is a child tread of $T$, the two colourings agree on
|
||||
$
|
||||
V(\mathsf{M}(T)) \cap V(\mathsf{M}(T')).
|
||||
$
|
||||
\end{definition}
|
||||
|
||||
\begin{proposition}[Gluing criterion]
|
||||
\label{prop:gluing-criterion}
|
||||
The full medial graph $M(G)$ has a proper vertex $3$-colouring if and
|
||||
only if the tire tree $\mathcal{T}(G,S)$ admits a compatible family of
|
||||
medial tire colourings.
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
A proper vertex $3$-colouring of $M(G)$ restricts to a proper vertex
|
||||
$3$-colouring of every induced subgraph $\mathsf{M}(T)$, and these
|
||||
restrictions agree on overlaps.
|
||||
|
||||
Conversely, suppose a compatible family is given. Define a colour on
|
||||
each vertex $m_e$ of $M(G)$ by choosing any tread $T$ with
|
||||
$m_e \in V(\mathsf{M}(T))$ and setting
|
||||
$\varphi(m_e)=\varphi_T(m_e)$. Compatibility makes this independent of
|
||||
the choice of $T$. Every medial edge of $M(G)$ is drawn in a corner of
|
||||
some bounded triangular face of $G$ or along the outer boundary
|
||||
interface. The relevant incident primal edges lie together in the
|
||||
closure of a single tire tread or in a shared boundary interface, where
|
||||
properness is already enforced by one of the local colourings. Hence
|
||||
$\varphi$ is a proper vertex $3$-colouring of $M(G)$.
|
||||
\end{proof}
|
||||
|
||||
\section{A medial pigeonhole programme}
|
||||
|
||||
The restriction relation $R_T$ records exactly the local information
|
||||
needed to pass a medial $3$-colouring through a tire. In a nested
|
||||
chain
|
||||
\[
|
||||
T_0 \supset T_1 \supset \cdots \supset T_k,
|
||||
\]
|
||||
the outer boundary state of $T_{i+1}$ must match an inner boundary
|
||||
state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
|
||||
this framework would follow from a structural reason that these
|
||||
restriction sets cannot remain mutually disjoint along every branch of
|
||||
the tire tree.
|
||||
|
||||
\begin{definition}[Medial boundary state]
|
||||
\label{def:medial-boundary-state}
|
||||
A \emph{medial boundary state} on a boundary set
|
||||
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
|
||||
subgraph induced by that boundary set, considered up to permutation of
|
||||
the three colours and the dihedral symmetries of the boundary walk
|
||||
when that boundary is a cycle.
|
||||
\end{definition}
|
||||
|
||||
\begin{conjecture}[Medial chain-pigeonhole principle]
|
||||
\label{conj:medial-chain-pigeonhole}
|
||||
There is a function $N(k)$ such that the following holds. Let
|
||||
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
|
||||
tire treads whose relevant boundary medial walks have length at most
|
||||
$k$. Then two adjacent restriction relations in the chain have
|
||||
compatible medial boundary states after colour permutation and boundary
|
||||
symmetry. Equivalently, the chain contains a local gluing step that
|
||||
cannot be obstructed by disjoint proper vertex $3$-colouring
|
||||
restrictions.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{conjecture}[Medial tire route to the Four Colour Theorem]
|
||||
\label{conj:medial-route-fct}
|
||||
For every plane triangulation $G$ and every level source $S$, the
|
||||
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
|
||||
compatible selection of boundary states across the tire tree. Hence
|
||||
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
|
||||
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{remark}
|
||||
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
|
||||
the Four Colour Theorem when combined with Tait's correspondence. The
|
||||
point of the formulation is not to weaken the target theorem, but to
|
||||
move the obstruction into finite boundary-state restrictions carried by
|
||||
annular medial tire pieces.
|
||||
\end{remark}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{bauerfeld-nested-tire-decompositions}
|
||||
E.~Bauerfeld,
|
||||
\emph{Nested Tire Decompositions of Plane Triangulations},
|
||||
manuscript (math-research repository), 2026.
|
||||
|
||||
\bibitem{tait-original}
|
||||
P.~G. Tait,
|
||||
\emph{Remarks on the colourings of maps},
|
||||
Proceedings of the Royal Society of Edinburgh \textbf{10} (1880),
|
||||
729--729.
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
@@ -8,50 +8,48 @@
|
||||
\citation{dvorak-lidicky-cones}
|
||||
\citation{heesch-untersuchungen}
|
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|
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|
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|
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\citation{robertson-sanders-seymour-thomas}
|
||||
\newlabel{def:dual}{{1.3}{2}}
|
||||
\newlabel{def:dual-depth}{{1.4}{2}}
|
||||
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|
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|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:dual-depth}{{1}{3}}
|
||||
\newlabel{def:tire-graph}{{1.6}{3}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:tire-example}{{2}{4}}
|
||||
\newlabel{def:medial-tire-graph}{{1.7}{4}}
|
||||
\newlabel{thm:annular-medial-colour-bound}{{1.8}{4}}
|
||||
\newlabel{rem:tire-counts}{{1.8}{4}}
|
||||
\newlabel{prop:no-level-d-pinch}{{1.9}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The medial tire graph for the tire in Figure\nonbreakingspace 2\hbox {}. The chord of $O$ is drawn faintly and omitted before taking the medial graph; medial edges between consecutive outer-boundary edges or consecutive inner-boundary edges are also omitted. Each medial vertex is placed at the midpoint of its corresponding retained tire edge.}}{5}{}\protected@file@percent }
|
||||
\newlabel{fig:medial-tire-example}{{3}{5}}
|
||||
\newlabel{rem:tire-counts}{{1.9}{5}}
|
||||
\newlabel{prop:no-level-d-pinch}{{1.10}{6}}
|
||||
\newlabel{lem:tire-component}{{1.11}{6}}
|
||||
\citation{bauerfeld-depth}
|
||||
\newlabel{lem:tire-component}{{1.10}{6}}
|
||||
\citation{bauerfeld-depth}
|
||||
\newlabel{thm:tread-partition}{{1.12}{8}}
|
||||
\newlabel{rem:tire-component-degenerate}{{1.13}{8}}
|
||||
\newlabel{rem:tire-no-extra-hypotheses}{{1.14}{8}}
|
||||
\newlabel{thm:inner-dual-outerplanar}{{1.15}{9}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{10}{}\protected@file@percent }
|
||||
\newlabel{fig:inner-dual-disk-case}{{4}{10}}
|
||||
\newlabel{thm:tread-partition}{{1.11}{7}}
|
||||
\newlabel{rem:tire-component-degenerate}{{1.12}{8}}
|
||||
\newlabel{rem:tire-no-extra-hypotheses}{{1.13}{8}}
|
||||
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|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{9}{}\protected@file@percent }
|
||||
\newlabel{fig:inner-dual-disk-case}{{4}{9}}
|
||||
\citation{bauerfeld-nested-tire-duals}
|
||||
\citation{bauerfeld-nested-tire-duals}
|
||||
\newlabel{rem:hamilton-cycle-spoke-only}{{1.15}{10}}
|
||||
\newlabel{rem:bridge-case-theta}{{1.16}{10}}
|
||||
\newlabel{thm:tread-tree}{{1.17}{10}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong \Theta (1, 7, 7)$ is outerplanar.}}{11}{}\protected@file@percent }
|
||||
\newlabel{fig:inner-dual-annulus-case}{{5}{11}}
|
||||
\newlabel{rem:hamilton-cycle-spoke-only}{{1.16}{11}}
|
||||
\newlabel{rem:bridge-case-theta}{{1.17}{11}}
|
||||
\newlabel{thm:tread-tree}{{1.18}{12}}
|
||||
\newlabel{rem:tree-multiple-children}{{1.19}{13}}
|
||||
\newlabel{thm:tire-tree-decomposition}{{1.20}{13}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
|
||||
\newlabel{fig:tire-tree-decomposition}{{6}{15}}
|
||||
\newlabel{rem:tree-coloring-factorisation}{{1.21}{15}}
|
||||
\newlabel{rem:tree-multiple-children}{{1.18}{12}}
|
||||
\newlabel{thm:tire-tree-decomposition}{{1.19}{12}}
|
||||
\bibcite{tait-original}{1}
|
||||
\bibcite{bauerfeld-depth}{2}
|
||||
\bibcite{bauerfeld-nested-tire-duals}{3}
|
||||
\bibcite{birkhoff-reducibility}{4}
|
||||
\bibcite{birkhoff-lewis-chromatic}{5}
|
||||
\bibcite{tutte-four-colour-conjecture}{6}
|
||||
\newlabel{rem:tree-coloring-factorisation}{{1.20}{14}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
|
||||
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|
||||
\bibcite{tutte-chromatic-sums-1973}{8}
|
||||
\bibcite{heesch-untersuchungen}{9}
|
||||
@@ -62,5 +60,6 @@
|
||||
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||||
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|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.19\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
|
||||
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|
||||
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Binary file not shown.
@@ -52,9 +52,8 @@ $G$ induces a BFS layering of $G$ and endows the inner planar dual
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$G'$ with a \emph{dual depth} grading. The basic object of study is
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the \emph{tire graph} $T$ --- a plane graph whose outer and inner
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boundaries bound a closed planar region, the \emph{tire tread} $R$,
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triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$. We define
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medial tire graphs and prove a basic colour-count bound for their
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annular medial cycle. Our main structural results are the
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triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$. Our main
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structural results are the
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\emph{tire-component lemma}, the \emph{tire-tread partition theorem},
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and the rooted \emph{tire-tree decomposition}, which together organize
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the bounded faces of $G$ into nested tire treads.
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@@ -283,52 +282,6 @@ corresponding retained tire edge.}
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\label{fig:medial-tire-example}
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\end{figure}
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\begin{theorem}[Annular medial colour bound]
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\label{thm:annular-medial-colour-bound}
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Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph with
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non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
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Let $A(T)$ be the subgraph of $M_{\mathrm{tire}}(T)$ induced by the
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annular medial vertices. For a graph $H$, write
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$\operatorname{Col}_3(H)$ for the set of proper $3$-vertex-colourings
|
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of $H$. Then $A(T)$ is a cycle and
|
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\[
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|\operatorname{Col}_3(M_{\mathrm{tire}}(T))|
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\;\leq\; |\operatorname{Col}_3(A(T))|.
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||||
\]
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\end{theorem}
|
||||
|
||||
\begin{proof}
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||||
Since the tread is a triangulated annulus with no vertices in its
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interior, each annular face has exactly one boundary edge, lying either
|
||||
on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular
|
||||
edges. As the annular faces are traversed cyclically around the tread,
|
||||
consecutive faces share one annular edge. Equivalently, the annular
|
||||
edges occur in a cyclic order in which each annular face contains two
|
||||
consecutive annular edges. Hence the subgraph of
|
||||
$M_{\mathrm{tire}}(T)$ induced by the annular medial vertices is a
|
||||
cycle.
|
||||
|
||||
Consider the restriction map from proper $3$-colourings of
|
||||
$M_{\mathrm{tire}}(T)$ to colourings of this annular medial cycle
|
||||
$A(T)$. We claim that this map is injective. Let $x$ be a
|
||||
non-annular medial vertex. Then $x$ corresponds to an edge of
|
||||
$B_{\mathrm{out}}$ or $B_{\mathrm{in}}$, since the chords of $O$ were
|
||||
omitted before forming $M_{\mathrm{tire}}(T)$. This boundary edge is
|
||||
incident to a unique annular face of $T^{\circ}$, and the other two
|
||||
edges of that face are annular edges. Therefore $x$ is adjacent in
|
||||
$M_{\mathrm{tire}}(T)$ to the two annular medial vertices corresponding
|
||||
to those two annular edges.
|
||||
|
||||
Those two annular medial vertices are adjacent to each other, because
|
||||
their annular edges are consecutive on the same triangular annular
|
||||
face. In any proper $3$-colouring they therefore receive two distinct
|
||||
colours, and $x$ is forced to receive the remaining third colour.
|
||||
Thus every non-annular medial vertex has its colour uniquely determined
|
||||
by the colouring of $A(T)$. Two colourings of $M_{\mathrm{tire}}(T)$
|
||||
with the same restriction to $A(T)$ are identical, so the restriction
|
||||
map is injective. The stated inequality follows.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:tire-counts}
|
||||
Let $\mu = |V(B_{\mathrm{out}})|$ and $\nu = |V(B_{\mathrm{in}})|$. By
|
||||
|
||||
Reference in New Issue
Block a user