Add Level Switching paper with surface-switch framework
Defines level cycles, edge switches, surface switches, and facial depth on level components of plane triangulations. Proves outerplanarity of level components and a depth-descent lemma. Introduces balanced surface switches and proves they remove a depth-d level cycle while creating 1-2 new depth-(d-1) cycles. Documents the 9-vertex counterexample where no balanced switch exists and sketches preprocessing toward balancedness, leaving general termination open. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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"""9-vertex L_k where the unique depth-1 face has NO balanced surface switch.
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Outer cycle: 0..8. Triangulated with chords 0-2, 0-3, 3-5, 3-6, 0-6, 6-8.
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Central triangle F = (0,3,6) has depth 1; its three neighbours
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(0,2,3), (3,5,6), (6,8,0) are all depth 0 but each has only ONE
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outer-cycle edge (not two), so none is an "ear" of F.
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For d = 1, balancedness requires F' to be an ear of uv (both non-uv
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edges on the outer cycle). No neighbour of F qualifies.
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"""
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import os
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import math
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import networkx as nx
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
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n = 9
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POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
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math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
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OUTER_EDGES = [(i, (i + 1) % n) for i in range(n)]
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CHORDS = [(0, 2), (0, 3), (3, 5), (3, 6), (0, 6), (6, 8)]
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FACES = [
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(0, 1, 2), # ear
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(0, 2, 3), # 1 outer edge, depth 0
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(3, 4, 5), # ear
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(3, 5, 6), # 1 outer edge, depth 0
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(6, 7, 8), # ear
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(6, 8, 0), # 1 outer edge, depth 0
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(0, 3, 6), # central, depth 1 -- the troublemaker
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]
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def face_edges(f):
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return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
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frozenset((f[0], f[2]))}
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outer_set = {frozenset(e) for e in OUTER_EDGES}
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D = nx.Graph()
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D.add_nodes_from(range(len(FACES)))
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for i, fi in enumerate(FACES):
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for j, fj in enumerate(FACES):
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if i < j and face_edges(fi) & face_edges(fj):
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D.add_edge(i, j)
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B = [i for i, f in enumerate(FACES)
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if len(face_edges(f) & outer_set) >= 1]
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depth = {i: min(nx.shortest_path_length(D, i, b) for b in B)
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for i in range(len(FACES))}
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palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
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edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
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fig, ax = plt.subplots(figsize=(7, 7))
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for i, f in enumerate(FACES):
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d = depth[i]
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poly = Polygon([POS[v] for v in f], closed=True,
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facecolor=palette[d], edgecolor=edge_pal[d],
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linewidth=1.6, alpha=0.7, zorder=0)
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ax.add_patch(poly)
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cx = sum(POS[v][0] for v in f) / 3
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cy = sum(POS[v][1] for v in f) / 3
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ax.text(cx, cy, rf'$\mathrm{{depth}}={d}$',
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ha='center', va='center', fontsize=10,
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color=edge_pal[d], fontweight='bold')
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# Mark the three "bad" chord edges (would-be-switched edges of F that
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# fail balancedness because the chord side has no outer-cycle edge to
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# pair with).
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F_edges = [(0, 3), (3, 6), (0, 6)]
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for (a, b) in OUTER_EDGES + CHORDS:
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color = '#333'; lw = 1.2
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if (a, b) in F_edges or (b, a) in F_edges:
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color = '#dc2626'; lw = 3.0
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ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
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color=color, linewidth=lw, zorder=1)
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for i, (x, y) in POS.items():
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ax.scatter([x], [y], s=300, c='#1f2937', edgecolors='black',
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linewidths=1.0, zorder=2)
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ax.text(x, y, str(i), ha='center', va='center',
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fontsize=10, color='white', fontweight='bold', zorder=3)
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ax.set_aspect('equal'); ax.axis('off')
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ax.set_xlim(-1.3, 1.3); ax.set_ylim(-1.3, 1.3)
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ax.set_title('Depth-1 face with no balanced surface switch',
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fontsize=12)
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_no_balanced_switch.png')
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fig.savefig(out, dpi=180, bbox_inches='tight')
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plt.close(fig)
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print(f'wrote {out}')
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for i, f in enumerate(FACES):
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print(f' {f} -> depth {depth[i]}')
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"""Counterexample showing that a surface switch on the edge between a
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depth-d face F and a depth-(d-1) face F' can create a new face of depth
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d (not d-1) when the depth-0 neighbor of F' lies on only one side of
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the shared edge.
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14-vertex maximal outerplanar L_k. Outer cycle order:
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u, p1, p2, p3, x, q1, v, b1, b2, b3, w, a1, a2, a3 -> u
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Central triangle F = (u, v, w) has depth 2.
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F' = (u, v, x) has depth 1 (its depth-0 neighbor is the q1-ear on the
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v-side of x; its u-side neighbor A_ux is depth 1).
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A_uw = (u, a2, w), A_vw = (v, b2, w) are both depth 1.
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Surface switch on uv: flip uv -> wx. New faces are
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A = (u, w, x) and B = (v, w, x).
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B inherits the depth-0 q1-ear, so depth(B) = 1.
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A's neighbors are A_uw (1), A_ux (1), B (1), so depth(A) = 2 = d. BAD.
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"""
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import os
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import math
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import networkx as nx
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
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OUTER = ['u', 'p1', 'p2', 'p3', 'x', 'q1', 'v',
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'b1', 'b2', 'b3', 'w', 'a1', 'a2', 'a3']
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n = len(OUTER)
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POS = {}
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for i, name in enumerate(OUTER):
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a = math.radians(90 - i * (360 / n))
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POS[name] = (math.cos(a), math.sin(a))
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OUTER_EDGES = [(OUTER[i], OUTER[(i + 1) % n]) for i in range(n)]
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CHORDS_BEFORE = [
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('u', 'v'), ('u', 'w'), ('v', 'w'), # central F = (u,v,w)
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('u', 'x'), ('v', 'x'), # F' = (u,v,x)
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('u', 'a2'), ('a2', 'w'), # A_uw = (u,a2,w)
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('v', 'b2'), ('b2', 'w'), # A_vw = (v,b2,w)
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('u', 'p2'), ('p2', 'x'), # A_ux = (u,p2,x)
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]
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CHORDS_AFTER = [c for c in CHORDS_BEFORE if set(c) != {'u', 'v'}] + [('w', 'x')]
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FACES_BEFORE = [
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('u', 'v', 'w'), # F (depth 2 -- bad)
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('u', 'v', 'x'), # F' (depth 1)
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('u', 'a2', 'w'), # A_uw (depth 1)
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('v', 'b2', 'w'), # A_vw (depth 1)
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('u', 'p2', 'x'), # A_ux (depth 1)
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('v', 'q1', 'x'), # A_vx (depth 0)
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('u', 'a1', 'a2'), # depth 0
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('a2', 'a3', 'w'), # depth 0
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('v', 'b1', 'b2'), # depth 0
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('b2', 'b3', 'w'), # depth 0
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('u', 'p1', 'p2'), # depth 0
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('p2', 'p3', 'x'), # depth 0
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]
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FACES_AFTER = [
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('u', 'w', 'x'), # A (depth 2 -- still bad!)
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('v', 'w', 'x'), # B (depth 1)
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('u', 'a2', 'w'),
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('v', 'b2', 'w'),
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('u', 'p2', 'x'),
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('v', 'q1', 'x'),
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('u', 'a1', 'a2'),
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('a2', 'a3', 'w'),
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('v', 'b1', 'b2'),
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('b2', 'b3', 'w'),
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('u', 'p1', 'p2'),
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('p2', 'p3', 'x'),
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]
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def compute_depths(faces, chords):
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"""Compute facial depth for each face using threshold-1 definition."""
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outer_set = {frozenset(e) for e in OUTER_EDGES}
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def face_edges(f):
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return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
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frozenset((f[0], f[2]))}
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# Build inner-face dual
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D = nx.Graph()
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D.add_nodes_from(range(len(faces)))
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for i, fi in enumerate(faces):
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for j, fj in enumerate(faces):
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if i < j and face_edges(fi) & face_edges(fj):
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D.add_edge(i, j)
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B = [i for i, f in enumerate(faces)
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if len(face_edges(f) & outer_set) >= 1]
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depth = {}
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for i in range(len(faces)):
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if not B:
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depth[i] = float('inf')
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continue
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depth[i] = min(nx.shortest_path_length(D, i, b) for b in B)
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return depth
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def draw_panel(ax, faces, chords, depth, title, highlight_edge=None,
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highlight_face=None):
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palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
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edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
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for i, f in enumerate(faces):
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d = depth[i]
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face_color = palette.get(d, '#ddd')
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face_edge = edge_pal.get(d, '#333')
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lw = 1.4
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if highlight_face is not None and i == highlight_face:
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face_edge = '#7c2d12'
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lw = 3.0
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poly = Polygon([POS[v] for v in f], closed=True,
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facecolor=face_color, edgecolor=face_edge,
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linewidth=lw, alpha=0.7, zorder=0)
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ax.add_patch(poly)
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cx = sum(POS[v][0] for v in f) / 3
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cy = sum(POS[v][1] for v in f) / 3
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ax.text(cx, cy, str(d), ha='center', va='center', fontsize=11,
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color=edge_pal.get(d, '#333'), fontweight='bold')
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# Draw edges
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all_edges = OUTER_EDGES + list(chords)
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for (a, b) in all_edges:
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color = '#333'; lw = 1.2
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if highlight_edge is not None and {a, b} == set(highlight_edge):
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color = '#dc2626'; lw = 3.5
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elif (a, b) == ('w', 'x') or (a, b) == ('x', 'w'):
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color = '#16a34a'; lw = 3.0
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ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
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color=color, linewidth=lw, zorder=1)
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# Draw vertices
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for name, (x, y) in POS.items():
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ax.scatter([x], [y], s=260, c='#1f2937', edgecolors='black',
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linewidths=0.8, zorder=2)
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ax.text(x, y, name, ha='center', va='center',
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fontsize=8, color='white', fontweight='bold', zorder=3)
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ax.set_aspect('equal'); ax.axis('off')
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ax.set_xlim(-1.35, 1.35); ax.set_ylim(-1.35, 1.35)
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ax.set_title(title, fontsize=12)
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def main():
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depth_before = compute_depths(FACES_BEFORE, CHORDS_BEFORE)
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depth_after = compute_depths(FACES_AFTER, CHORDS_AFTER)
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print("BEFORE:")
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for i, f in enumerate(FACES_BEFORE):
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print(f" {f} -> depth {depth_before[i]}")
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print("AFTER:")
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for i, f in enumerate(FACES_AFTER):
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print(f" {f} -> depth {depth_after[i]}")
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fig, axes = plt.subplots(1, 2, figsize=(15, 7.5))
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draw_panel(axes[0], FACES_BEFORE, CHORDS_BEFORE, depth_before,
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r'Before: $F=(u,v,w)$ depth 2, $F\'=(u,v,x)$ depth 1',
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highlight_edge=('u', 'v'), highlight_face=0)
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draw_panel(axes[1], FACES_AFTER, CHORDS_AFTER, depth_after,
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r'After surface switch on $uv$: $A=(u,w,x)$ still depth 2!',
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highlight_face=0)
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_counterexample_surface_switch.png')
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fig.savefig(out, dpi=180, bbox_inches='tight')
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plt.close(fig)
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print(f'wrote {out}')
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if __name__ == '__main__':
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main()
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@@ -0,0 +1,370 @@
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"""Generate the three definition figures for the Level Switching paper.
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Uses a stacked 7-vertex triangulation T:
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outer triangle {0,1,2}, inner vertex 3 connected to all three,
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then vertices 4,5,6 inserted into faces (1,2,3),(0,2,3),(0,1,3).
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"""
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import os
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import networkx as nx
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
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# Vertex positions (hand-placed for a clean planar drawing).
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POS = {
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0: (-1.5, -0.9),
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1: (1.5, -0.9),
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2: (0.0, 1.6),
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3: (0.0, 0.0),
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4: (0.55, 0.2), # in face (1,2,3)
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5: (-0.55, 0.2), # in face (0,2,3)
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6: (0.0, -0.55), # in face (0,1,3)
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}
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EDGES = [
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(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), # K_4
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(4, 1), (4, 2), (4, 3), # stack in (1,2,3)
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(5, 0), (5, 2), (5, 3), # stack in (0,2,3)
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(6, 0), (6, 1), (6, 3), # stack in (0,1,3)
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]
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def make_graph():
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G = nx.Graph()
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G.add_nodes_from(POS.keys())
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G.add_edges_from(EDGES)
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return G
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def draw_base(ax, G, node_colors, node_size=520, font_color='white',
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edge_color='#555', edge_width=1.4):
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nx.draw_networkx_edges(G, POS, ax=ax, edge_color=edge_color, width=edge_width)
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nx.draw_networkx_nodes(G, POS, ax=ax, node_color=node_colors,
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node_size=node_size, edgecolors='black', linewidths=1.0)
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nx.draw_networkx_labels(G, POS, ax=ax, font_color=font_color,
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font_size=11, font_weight='bold')
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ax.set_aspect('equal')
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ax.axis('off')
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# ---------------------------------------------------------------------------
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# Figure 1: Level source (face source vs. degree-3 vertex source)
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# ---------------------------------------------------------------------------
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def fig_level_source():
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G = make_graph()
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fig, axes = plt.subplots(1, 2, figsize=(10, 5))
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# Panel A: face source S = {0,1,2}
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ax = axes[0]
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face_S = {0, 1, 2}
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colors = ['#ef4444' if v in face_S else '#cbd5e1' for v in G.nodes()]
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# Highlight the source face
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tri = Polygon([POS[v] for v in [0, 1, 2]], closed=True,
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facecolor='#fecaca', edgecolor='#ef4444', linewidth=2.0,
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alpha=0.45, zorder=0)
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ax.add_patch(tri)
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draw_base(ax, G, colors)
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ax.set_title(r'Face source $S = \{0,1,2\}$', fontsize=12)
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# Panel B: degree-3 vertex source S = {4}
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ax = axes[1]
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vert_S = {4}
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colors = ['#ef4444' if v in vert_S else '#cbd5e1' for v in G.nodes()]
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draw_base(ax, G, colors)
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ax.set_title(r'Degree-3 vertex source $S = \{4\}$', fontsize=12)
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_level_source.png')
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fig.savefig(out, dpi=200, bbox_inches='tight')
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plt.close(fig)
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print(f'wrote {out}')
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# ---------------------------------------------------------------------------
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# Figure 2: Levels (BFS distance from a source)
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# ---------------------------------------------------------------------------
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def fig_levels():
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G = make_graph()
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source = 4 # degree-3 vertex source
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levels = nx.single_source_shortest_path_length(G, source)
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# Color by level
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palette = {0: '#ef4444', 1: '#f59e0b', 2: '#3b82f6'}
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colors = [palette[levels[v]] for v in G.nodes()]
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# Labels = level numbers
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labels = {v: rf'$\ell={levels[v]}$' for v in G.nodes()}
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fig, ax = plt.subplots(figsize=(6.5, 5.5))
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nx.draw_networkx_edges(G, POS, ax=ax, edge_color='#555', width=1.4)
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nx.draw_networkx_nodes(G, POS, ax=ax, node_color=colors,
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node_size=720, edgecolors='black', linewidths=1.0)
|
||||
# Draw vertex id slightly above, level label inside
|
||||
for v, (x, y) in POS.items():
|
||||
ax.text(x, y, str(v), ha='center', va='center',
|
||||
fontsize=10, fontweight='bold', color='white')
|
||||
ax.text(x + 0.18, y + 0.18, rf'$\ell={levels[v]}$',
|
||||
fontsize=10, color='black',
|
||||
bbox=dict(boxstyle='round,pad=0.15',
|
||||
facecolor='white', edgecolor='#999', linewidth=0.6))
|
||||
ax.set_aspect('equal')
|
||||
ax.axis('off')
|
||||
ax.set_title(r'Levels $\ell_G(v)$ from source $S=\{4\}$', fontsize=12)
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_levels.png')
|
||||
fig.savefig(out, dpi=200, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
|
||||
|
||||
# ---------------------------------------------------------------------------
|
||||
# Figure 3: Parity subgraph (even and odd induced subgraphs)
|
||||
# ---------------------------------------------------------------------------
|
||||
def fig_parity_subgraph():
|
||||
G = make_graph()
|
||||
source = 4
|
||||
levels = nx.single_source_shortest_path_length(G, source)
|
||||
parity = {v: levels[v] % 2 for v in G.nodes()}
|
||||
|
||||
even = [v for v in G.nodes() if parity[v] == 0]
|
||||
odd = [v for v in G.nodes() if parity[v] == 1]
|
||||
even_color = '#3b82f6' # blue
|
||||
odd_color = '#f59e0b' # orange
|
||||
|
||||
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
|
||||
|
||||
# Panel A: full triangulation, vertices coloured by parity
|
||||
ax = axes[0]
|
||||
colors = [even_color if parity[v] == 0 else odd_color for v in G.nodes()]
|
||||
draw_base(ax, G, colors)
|
||||
ax.set_title(r"$G'$ with vertices coloured by $\ell_G$ mod 2", fontsize=12)
|
||||
|
||||
# Panel B: even parity subgraph (induced on even vertices)
|
||||
ax = axes[1]
|
||||
# Draw all edges faintly, then the induced subgraph in colour
|
||||
nx.draw_networkx_edges(G, POS, ax=ax, edge_color='#ddd', width=1.0)
|
||||
even_sub = G.subgraph(even)
|
||||
nx.draw_networkx_edges(even_sub, POS, ax=ax, edge_color=even_color, width=2.4)
|
||||
node_colors = [even_color if v in even else '#e5e7eb' for v in G.nodes()]
|
||||
nx.draw_networkx_nodes(G, POS, ax=ax, node_color=node_colors,
|
||||
node_size=520, edgecolors='black', linewidths=1.0)
|
||||
nx.draw_networkx_labels(G, POS, ax=ax,
|
||||
font_color='white', font_size=11, font_weight='bold')
|
||||
ax.set_aspect('equal')
|
||||
ax.axis('off')
|
||||
ax.set_title(r"Even parity subgraph $E_{G,S}(G')$", fontsize=12)
|
||||
|
||||
# Panel C: odd parity subgraph
|
||||
ax = axes[2]
|
||||
nx.draw_networkx_edges(G, POS, ax=ax, edge_color='#ddd', width=1.0)
|
||||
odd_sub = G.subgraph(odd)
|
||||
nx.draw_networkx_edges(odd_sub, POS, ax=ax, edge_color=odd_color, width=2.4)
|
||||
node_colors = [odd_color if v in odd else '#e5e7eb' for v in G.nodes()]
|
||||
nx.draw_networkx_nodes(G, POS, ax=ax, node_color=node_colors,
|
||||
node_size=520, edgecolors='black', linewidths=1.0)
|
||||
nx.draw_networkx_labels(G, POS, ax=ax,
|
||||
font_color='white', font_size=11, font_weight='bold')
|
||||
ax.set_aspect('equal')
|
||||
ax.axis('off')
|
||||
ax.set_title(r"Odd parity subgraph $O_{G,S}(G')$", fontsize=12)
|
||||
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_parity_subgraph.png')
|
||||
fig.savefig(out, dpi=200, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
|
||||
|
||||
# ---------------------------------------------------------------------------
|
||||
# Figure: Level cycle (simple cycle within a single level)
|
||||
# ---------------------------------------------------------------------------
|
||||
def fig_level_cycle():
|
||||
G = make_graph()
|
||||
source = 4
|
||||
levels = nx.single_source_shortest_path_length(G, source)
|
||||
|
||||
palette = {0: '#ef4444', 1: '#f59e0b', 2: '#3b82f6'}
|
||||
colors = [palette[levels[v]] for v in G.nodes()]
|
||||
|
||||
# Level cycle: 1-2-3-1 lies entirely in L_1
|
||||
cycle_edges = [(1, 2), (2, 3), (1, 3)]
|
||||
|
||||
fig, ax = plt.subplots(figsize=(6.5, 5.5))
|
||||
nx.draw_networkx_edges(G, POS, ax=ax, edge_color='#bbb', width=1.2)
|
||||
nx.draw_networkx_edges(G, POS, edgelist=cycle_edges, ax=ax,
|
||||
edge_color='#dc2626', width=3.4)
|
||||
nx.draw_networkx_nodes(G, POS, ax=ax, node_color=colors,
|
||||
node_size=620, edgecolors='black', linewidths=1.0)
|
||||
nx.draw_networkx_labels(G, POS, ax=ax, font_color='white',
|
||||
font_size=11, font_weight='bold')
|
||||
# Annotate levels in small floating labels
|
||||
for v, (x, y) in POS.items():
|
||||
ax.text(x + 0.18, y + 0.18, rf'$\ell={levels[v]}$',
|
||||
fontsize=9, color='black',
|
||||
bbox=dict(boxstyle='round,pad=0.12',
|
||||
facecolor='white', edgecolor='#999', linewidth=0.5))
|
||||
ax.set_aspect('equal')
|
||||
ax.axis('off')
|
||||
ax.set_title(r'Level cycle in $L_1 = \{1,2,3\}$ (highlighted)', fontsize=12)
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_level_cycle.png')
|
||||
fig.savefig(out, dpi=200, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
|
||||
|
||||
# ---------------------------------------------------------------------------
|
||||
# Figure: Edge switch (flip on a level-cycle edge)
|
||||
# ---------------------------------------------------------------------------
|
||||
def fig_edge_switch():
|
||||
G = make_graph()
|
||||
source = 4
|
||||
levels = nx.single_source_shortest_path_length(G, source)
|
||||
palette = {0: '#ef4444', 1: '#f59e0b', 2: '#3b82f6'}
|
||||
colors = [palette[levels[v]] for v in G.nodes()]
|
||||
|
||||
# We switch edge (1,2), which lies in the L_1 cycle 1-2-3-1.
|
||||
# Its two adjacent faces in T are (0,1,2) and (1,2,4); the flip
|
||||
# removes 1-2 and adds 0-4.
|
||||
removed = (1, 2)
|
||||
added = (0, 4)
|
||||
|
||||
Gprime = G.copy()
|
||||
Gprime.remove_edge(*removed)
|
||||
Gprime.add_edge(*added)
|
||||
|
||||
fig, axes = plt.subplots(1, 2, figsize=(12, 5.5))
|
||||
|
||||
# Panel A: before — highlight the level-cycle edge to be switched
|
||||
ax = axes[0]
|
||||
other_edges = [e for e in G.edges() if set(e) != set(removed)]
|
||||
nx.draw_networkx_edges(G, POS, edgelist=other_edges, ax=ax,
|
||||
edge_color='#bbb', width=1.2)
|
||||
nx.draw_networkx_edges(G, POS, edgelist=[removed], ax=ax,
|
||||
edge_color='#dc2626', width=3.4)
|
||||
nx.draw_networkx_nodes(G, POS, ax=ax, node_color=colors,
|
||||
node_size=560, edgecolors='black', linewidths=1.0)
|
||||
nx.draw_networkx_labels(G, POS, ax=ax, font_color='white',
|
||||
font_size=11, font_weight='bold')
|
||||
ax.set_aspect('equal'); ax.axis('off')
|
||||
ax.set_title(r'Before: edge $1\!-\!2$ lies on the $L_1$ cycle',
|
||||
fontsize=12)
|
||||
|
||||
# Panel B: after — the new edge highlighted in green
|
||||
ax = axes[1]
|
||||
other_edges = [e for e in Gprime.edges() if set(e) != set(added)]
|
||||
nx.draw_networkx_edges(Gprime, POS, edgelist=other_edges, ax=ax,
|
||||
edge_color='#bbb', width=1.2)
|
||||
nx.draw_networkx_edges(Gprime, POS, edgelist=[added], ax=ax,
|
||||
edge_color='#16a34a', width=3.4)
|
||||
nx.draw_networkx_nodes(Gprime, POS, ax=ax, node_color=colors,
|
||||
node_size=560, edgecolors='black', linewidths=1.0)
|
||||
nx.draw_networkx_labels(Gprime, POS, ax=ax, font_color='white',
|
||||
font_size=11, font_weight='bold')
|
||||
ax.set_aspect('equal'); ax.axis('off')
|
||||
ax.set_title(r'After: $1\!-\!2$ replaced by $0\!-\!4$',
|
||||
fontsize=12)
|
||||
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_edge_switch.png')
|
||||
fig.savefig(out, dpi=200, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
|
||||
|
||||
# ---------------------------------------------------------------------------
|
||||
# Figure: Facial depth (depths in an outerplanar L_k)
|
||||
# ---------------------------------------------------------------------------
|
||||
def fig_facial_depth():
|
||||
import math
|
||||
# 12-vertex maximal outerplanar graph with 3-fold symmetry.
|
||||
# Central triangle (0,4,8); three "in-between" triangles (0,2,4),
|
||||
# (4,6,8), (0,8,10) sit between the central triangle and the
|
||||
# outer "ears" (0,1,2), (2,3,4), (4,5,6), (6,7,8), (8,9,10),
|
||||
# (10,11,0).
|
||||
n = 12
|
||||
pos = {}
|
||||
for i in range(n):
|
||||
a = math.radians(90 - i * (360 / n))
|
||||
pos[i] = (math.cos(a), math.sin(a))
|
||||
|
||||
outer_edges = [(i, (i + 1) % n) for i in range(n)]
|
||||
diagonals = [(0, 2), (2, 4), (4, 6), (6, 8), (8, 10), (10, 0), # "short" chords
|
||||
(0, 4), (4, 8), (0, 8)] # central triangle
|
||||
L = nx.Graph()
|
||||
L.add_nodes_from(pos)
|
||||
L.add_edges_from(outer_edges + diagonals)
|
||||
|
||||
inner_faces = [
|
||||
(0, 1, 2), (2, 3, 4), (4, 5, 6),
|
||||
(6, 7, 8), (8, 9, 10), (10, 11, 0), # 6 outer "ears"
|
||||
(0, 2, 4), (4, 6, 8), (0, 8, 10), # 3 in-between
|
||||
(0, 4, 8), # central
|
||||
]
|
||||
|
||||
# Build dual graph on inner faces: edge iff faces share an edge
|
||||
def face_edges(f):
|
||||
a, b, c = f
|
||||
return {frozenset((a, b)), frozenset((b, c)), frozenset((a, c))}
|
||||
|
||||
outer_edge_set = {frozenset(e) for e in outer_edges}
|
||||
D = nx.Graph()
|
||||
D.add_nodes_from(range(len(inner_faces)))
|
||||
for i, fi in enumerate(inner_faces):
|
||||
for j, fj in enumerate(inner_faces):
|
||||
if i < j and face_edges(fi) & face_edges(fj):
|
||||
D.add_edge(i, j)
|
||||
|
||||
# Boundary set B: faces whose bounding level cycle has >= 1 outer-cycle edge
|
||||
B = [i for i, f in enumerate(inner_faces)
|
||||
if len(face_edges(f) & outer_edge_set) >= 1]
|
||||
|
||||
# Depth = min distance in D to any face in B
|
||||
depth = {}
|
||||
for i in range(len(inner_faces)):
|
||||
dists = [nx.shortest_path_length(D, i, b) for b in B]
|
||||
depth[i] = min(dists)
|
||||
|
||||
# Colour by depth
|
||||
depth_color = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
|
||||
depth_edge = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
|
||||
|
||||
fig, ax = plt.subplots(figsize=(7, 7))
|
||||
# Fill faces by depth
|
||||
for i, f in enumerate(inner_faces):
|
||||
poly = Polygon([pos[v] for v in f], closed=True,
|
||||
facecolor=depth_color[depth[i]],
|
||||
edgecolor=depth_edge[depth[i]],
|
||||
linewidth=1.5, alpha=0.7, zorder=0)
|
||||
ax.add_patch(poly)
|
||||
cx = sum(pos[v][0] for v in f) / 3
|
||||
cy = sum(pos[v][1] for v in f) / 3
|
||||
ax.text(cx, cy, rf'$\mathrm{{depth}}={depth[i]}$',
|
||||
ha='center', va='center', fontsize=10,
|
||||
color=depth_edge[depth[i]], fontweight='bold')
|
||||
|
||||
# Draw the graph on top
|
||||
nx.draw_networkx_edges(L, pos, ax=ax, edge_color='#333', width=1.4)
|
||||
nx.draw_networkx_nodes(L, pos, ax=ax, node_color='#1f2937',
|
||||
node_size=320, edgecolors='black', linewidths=1.0)
|
||||
nx.draw_networkx_labels(L, pos, ax=ax, font_color='white',
|
||||
font_size=10, font_weight='bold')
|
||||
|
||||
ax.set_aspect('equal'); ax.axis('off')
|
||||
ax.set_xlim(-1.3, 1.3); ax.set_ylim(-1.3, 1.3)
|
||||
ax.set_title(r'Facial depth in an outerplanar $L_k$', fontsize=12)
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_facial_depth.png')
|
||||
fig.savefig(out, dpi=200, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
fig_level_source()
|
||||
fig_levels()
|
||||
fig_level_cycle()
|
||||
fig_edge_switch()
|
||||
fig_parity_subgraph()
|
||||
fig_facial_depth()
|
||||
@@ -0,0 +1,118 @@
|
||||
"""Demonstrate the preprocessing strategy on the 9-vertex example.
|
||||
|
||||
Start: F = (0,3,6) at depth 1, no balanced surface switch exists
|
||||
(F has no edge of "span 1" -- no edge with a single outer-cycle
|
||||
vertex between the endpoints, hence no ear neighbour).
|
||||
|
||||
Step: perform the (unbalanced) surface switch on edge uv = 03, with
|
||||
F' = (0,2,3) and third vertex x = 2; in Case (ii) the flip removes 03
|
||||
and adds wx = 62.
|
||||
|
||||
Result: A = (0,2,6) at depth 1 has edge 02 at span 1, so the ear
|
||||
(0,1,2) is now a balanced-switch target.
|
||||
"""
|
||||
import os
|
||||
import math
|
||||
import networkx as nx
|
||||
import matplotlib.pyplot as plt
|
||||
from matplotlib.patches import Polygon
|
||||
|
||||
OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
|
||||
|
||||
n = 9
|
||||
POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
|
||||
math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
|
||||
OUTER_EDGES = [(i, (i + 1) % n) for i in range(n)]
|
||||
outer_set = {frozenset(e) for e in OUTER_EDGES}
|
||||
|
||||
|
||||
def face_edges(f):
|
||||
return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
|
||||
frozenset((f[0], f[2]))}
|
||||
|
||||
|
||||
def compute_depths(faces):
|
||||
D = nx.Graph()
|
||||
D.add_nodes_from(range(len(faces)))
|
||||
for i, fi in enumerate(faces):
|
||||
for j, fj in enumerate(faces):
|
||||
if i < j and face_edges(fi) & face_edges(fj):
|
||||
D.add_edge(i, j)
|
||||
B = [i for i, f in enumerate(faces)
|
||||
if len(face_edges(f) & outer_set) >= 1]
|
||||
return {i: min(nx.shortest_path_length(D, i, b) for b in B)
|
||||
for i in range(len(faces))}
|
||||
|
||||
|
||||
CHORDS_BEFORE = [(0, 2), (0, 3), (3, 5), (3, 6), (0, 6), (6, 8)]
|
||||
FACES_BEFORE = [
|
||||
(0, 1, 2), (0, 2, 3), (3, 4, 5), (3, 5, 6),
|
||||
(6, 7, 8), (6, 8, 0), (0, 3, 6),
|
||||
]
|
||||
|
||||
# After non-balanced switch on edge 03: remove 03, add 26
|
||||
CHORDS_AFTER = [c for c in CHORDS_BEFORE if set(c) != {0, 3}] + [(2, 6)]
|
||||
FACES_AFTER = [
|
||||
(0, 1, 2), (3, 4, 5), (3, 5, 6),
|
||||
(6, 7, 8), (6, 8, 0),
|
||||
(0, 2, 6), (2, 3, 6),
|
||||
]
|
||||
|
||||
|
||||
def draw(ax, faces, chords, depth, title, highlight_edges=None,
|
||||
green_edges=None):
|
||||
palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
|
||||
edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
|
||||
for i, f in enumerate(faces):
|
||||
d = depth[i]
|
||||
poly = Polygon([POS[v] for v in f], closed=True,
|
||||
facecolor=palette.get(d, '#ddd'),
|
||||
edgecolor=edge_pal.get(d, '#333'),
|
||||
linewidth=1.4, alpha=0.7, zorder=0)
|
||||
ax.add_patch(poly)
|
||||
cx = sum(POS[v][0] for v in f) / 3
|
||||
cy = sum(POS[v][1] for v in f) / 3
|
||||
ax.text(cx, cy, str(d), ha='center', va='center', fontsize=11,
|
||||
color=edge_pal.get(d, '#333'), fontweight='bold')
|
||||
for (a, b) in OUTER_EDGES + chords:
|
||||
color = '#333'; lw = 1.2
|
||||
if highlight_edges and ((a, b) in highlight_edges or
|
||||
(b, a) in highlight_edges):
|
||||
color = '#dc2626'; lw = 3.0
|
||||
if green_edges and ((a, b) in green_edges or
|
||||
(b, a) in green_edges):
|
||||
color = '#16a34a'; lw = 3.0
|
||||
ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
|
||||
color=color, linewidth=lw, zorder=1)
|
||||
for i, (x, y) in POS.items():
|
||||
ax.scatter([x], [y], s=270, c='#1f2937', edgecolors='black',
|
||||
linewidths=1.0, zorder=2)
|
||||
ax.text(x, y, str(i), ha='center', va='center',
|
||||
fontsize=9, color='white', fontweight='bold', zorder=3)
|
||||
ax.set_aspect('equal'); ax.axis('off')
|
||||
ax.set_xlim(-1.3, 1.3); ax.set_ylim(-1.3, 1.3)
|
||||
ax.set_title(title, fontsize=11)
|
||||
|
||||
|
||||
depth_before = compute_depths(FACES_BEFORE)
|
||||
depth_after = compute_depths(FACES_AFTER)
|
||||
print('BEFORE:')
|
||||
for i, f in enumerate(FACES_BEFORE):
|
||||
print(f' {f} -> depth {depth_before[i]}')
|
||||
print('AFTER:')
|
||||
for i, f in enumerate(FACES_AFTER):
|
||||
print(f' {f} -> depth {depth_after[i]}')
|
||||
|
||||
fig, axes = plt.subplots(1, 2, figsize=(14, 7))
|
||||
draw(axes[0], FACES_BEFORE, CHORDS_BEFORE, depth_before,
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'Before: F=(0,3,6) depth 1; spans (2,2,2) so no ear neighbour',
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highlight_edges=[(0, 3)])
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draw(axes[1], FACES_AFTER, CHORDS_AFTER, depth_after,
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'After non-balanced switch 03->26: A=(0,2,6) depth 1; edge 02 has span 1',
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green_edges=[(2, 6)], highlight_edges=[(0, 2)])
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_preprocessing.png')
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The two kinds of level source on a 7-vertex triangulation $T$ (K\textsubscript {4} with vertices $4,5,6$ stacked into the three interior faces). Left: the face source $S = \{0,1,2\}$ (level-0 vertices are the corners of the highlighted triangle). Right: the degree-$3$ vertex source $S = \{4\}$.}}{1}{figure.1}\protected@file@percent }
|
||||
\newlabel{fig:level-source}{{1}{1}{The two kinds of level source on a 7-vertex triangulation $T$ (K\textsubscript {4} with vertices $4,5,6$ stacked into the three interior faces). Left: the face source $S = \{0,1,2\}$ (level-0 vertices are the corners of the highlighted triangle). Right: the degree-$3$ vertex source $S = \{4\}$}{figure.1}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level.}}{2}{figure.2}\protected@file@percent }
|
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\newlabel{fig:levels}{{2}{2}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level}{figure.2}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A level cycle in the triangulation of Figure\nonbreakingspace \ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$.}}{2}{figure.3}\protected@file@percent }
|
||||
\newlabel{fig:level-cycle}{{3}{2}{A level cycle in the triangulation of Figure~\ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$}{figure.3}{}}
|
||||
\newlabel{def:edge-switch}{{2.4}{2}{Edge switch}{theorem.2.4}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces An edge switch on the level cycle of Figure\nonbreakingspace \ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$.}}{3}{figure.4}\protected@file@percent }
|
||||
\newlabel{fig:edge-switch}{{4}{3}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.4}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$.}}{3}{figure.5}\protected@file@percent }
|
||||
\newlabel{fig:parity-subgraph}{{5}{3}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.5}{}}
|
||||
\newlabel{def:facial-depth}{{2.6}{3}{Facial depth}{theorem.2.6}{}}
|
||||
\newlabel{def:surface-switch}{{2.7}{3}{Surface switch}{theorem.2.7}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Facial depths in a maximal outerplanar graph on $12$ vertices. The six green ear-triangles share an edge with the outer $12$-cycle and so lie in $\mathcal {B}$ (depth $0$). The three yellow ``in-between'' triangles $(0,2,4),(4,6,8),(0,8,10)$ have only diagonal edges but each is dual-adjacent to ears, giving them $\mathrm {depth} = 1$. The central triangle $(0,4,8)$ is also all-diagonal; its dual neighbours are the three depth-$1$ triangles, so it is isolated with $\mathrm {depth} = 2$.}}{4}{figure.6}\protected@file@percent }
|
||||
\newlabel{fig:facial-depth}{{6}{4}{Facial depths in a maximal outerplanar graph on $12$ vertices. The six green ear-triangles share an edge with the outer $12$-cycle and so lie in $\mathcal {B}$ (depth $0$). The three yellow ``in-between'' triangles $(0,2,4),(4,6,8),(0,8,10)$ have only diagonal edges but each is dual-adjacent to ears, giving them $\mathrm {depth} = 1$. The central triangle $(0,4,8)$ is also all-diagonal; its dual neighbours are the three depth-$1$ triangles, so it is isolated with $\mathrm {depth} = 2$}{figure.6}{}}
|
||||
\newlabel{def:balanced-surface-switch}{{2.8}{4}{Balanced surface switch}{theorem.2.8}{}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{4}{section.3}\protected@file@percent }
|
||||
\newlabel{sec:outerplanar-components}{{3}{4}{Outerplanarity of level components}{section.3}{}}
|
||||
\newlabel{thm:outerplanar-component}{{3.1}{4}{}{theorem.3.1}{}}
|
||||
\newlabel{lem:depth-descent}{{3.2}{5}{}{theorem.3.2}{}}
|
||||
\newlabel{prop:balanced-descent}{{3.3}{5}{}{theorem.3.3}{}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{When does a balanced surface switch exist?}}{6}{section*.1}\protected@file@percent }
|
||||
\newlabel{obs:span1-balanced-d1}{{3.4}{6}{}{theorem.3.4}{}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Preprocessing toward balanced switches}}{6}{section*.2}\protected@file@percent }
|
||||
\newlabel{ex:preprocessing}{{3.5}{6}{}{theorem.3.5}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces $9$-vertex maximal outerplanar $L_k$. $F = (0,3,6)$ has $\mathrm {depth} = 1$ and all three of its edges have span $2$, so none of $F$'s depth-$0$ neighbours is an ear. No balanced surface switch is available on $F$.}}{7}{figure.7}\protected@file@percent }
|
||||
\newlabel{fig:no-balanced}{{7}{7}{$9$-vertex maximal outerplanar $L_k$. $F = (0,3,6)$ has $\mathrm {depth} = 1$ and all three of its edges have span $2$, so none of $F$'s depth-$0$ neighbours is an ear. No balanced surface switch is available on $F$}{figure.7}{}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces One step of preprocessing on the $9$-vertex example. Left: $F = (0,3,6)$ has no edge of span $1$; the chosen surface-switch edge $uv = 03$ (red) is unbalanced. Right: after the switch $03 \DOTSB \mapstochar \rightarrow 26$ (green), the new depth-$1$ face $A = (0,2,6)$ has its edge $02$ (red) at span $1$, exposing the ear $(0,1,2)$ as a balanced surface-switch target.}}{7}{figure.8}\protected@file@percent }
|
||||
\newlabel{fig:preprocessing}{{8}{7}{One step of preprocessing on the $9$-vertex example. Left: $F = (0,3,6)$ has no edge of span $1$; the chosen surface-switch edge $uv = 03$ (red) is unbalanced. Right: after the switch $03 \mapsto 26$ (green), the new depth-$1$ face $A = (0,2,6)$ has its edge $02$ (red) at span $1$, exposing the ear $(0,1,2)$ as a balanced surface-switch target}{figure.8}{}}
|
||||
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|
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||||
\newlabel{q:preprocessing-terminates}{{3.6}{8}{}{theorem.3.6}{}}
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|
||||
\begin{definition}[Level source]
|
||||
A \emph{level source} of $G$ is either:
|
||||
\begin{itemize}
|
||||
\item a face $F$ of $G$ (all vertices of $F$ are level-0 sources), or
|
||||
\item a vertex $v$ of degree 3 (the singleton $\{v\}$ is a level-0 source).
|
||||
\end{itemize}
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{fig_level_source.png}
|
||||
\caption{The two kinds of level source on a 7-vertex triangulation $T$
|
||||
(K\textsubscript{4} with vertices $4,5,6$ stacked into the three
|
||||
interior faces). Left: the face source $S = \{0,1,2\}$
|
||||
(level-0 vertices are the corners of the highlighted triangle).
|
||||
Right: the degree-$3$ vertex source $S = \{4\}$.}
|
||||
\label{fig:level-source}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[Levels]
|
||||
Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is
|
||||
$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
|
||||
source vertex.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.55\textwidth]{fig_levels.png}
|
||||
\caption{BFS levels from the degree-$3$ vertex source $S = \{4\}$.
|
||||
The source is level $0$, its three neighbours are level $1$, and the
|
||||
remaining vertices are level $2$. Colour encodes the level.}
|
||||
\label{fig:levels}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[Level cycle]
|
||||
A \emph{level cycle} of $G$ (with respect to a level source $S$) is a
|
||||
simple cycle in $G$ all of whose vertices have the same level.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.55\textwidth]{fig_level_cycle.png}
|
||||
\caption{A level cycle in the triangulation of Figure~\ref{fig:levels}.
|
||||
The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all
|
||||
lie at level $1$, so it is a level cycle at level $1$.}
|
||||
\label{fig:level-cycle}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[Edge switch]
|
||||
\label{def:edge-switch}
|
||||
Let $G$ be a triangulation with level source $S$, and let $e = uv$ be an
|
||||
edge of a level cycle of $G$. The \emph{edge switch} at $e$ is the edge
|
||||
flip on $e$: writing $uvw$ and $uvx$ for the two triangular faces of $G$
|
||||
containing $e$, the edge $uv$ is removed and the edge $wx$ is added. As
|
||||
with any edge flip, the result is a triangulation on the same vertex set
|
||||
provided $w$ and $x$ are non-adjacent in $G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.95\textwidth]{fig_edge_switch.png}
|
||||
\caption{An edge switch on the level cycle of
|
||||
Figure~\ref{fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared
|
||||
by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes
|
||||
$1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex
|
||||
colours indicate the original levels in $G$.}
|
||||
\label{fig:edge-switch}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[Parity subgraph]
|
||||
Let $G$ be a triangulation with level source $S$, and let $G'$ be a triangulation
|
||||
on the same vertex set as $G$. The \emph{even parity subgraph} $E_{G,S}(G')$ is
|
||||
the subgraph of $G'$ induced by $\{v \in V : \ell_G(v) \equiv 0 \pmod 2\}$. The
|
||||
\emph{odd parity subgraph} is defined analogously for odd $\ell_G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{fig_parity_subgraph.png}
|
||||
\caption{Parity subgraphs of $G' = T$ with respect to the level structure of
|
||||
Figure~\ref{fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices
|
||||
coloured by $\ell_G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the
|
||||
even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only
|
||||
edges with both endpoints even appear. Right: the odd parity subgraph
|
||||
$O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows
|
||||
that $O_{G,S}(G')$ is not bipartite for this choice of $G'$.}
|
||||
\label{fig:parity-subgraph}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[Facial depth]
|
||||
\label{def:facial-depth}
|
||||
Let $L_k$ be drawn with the outerplanar embedding inherited from $\Pi_G$,
|
||||
let $D$ be the dual graph of this drawing with the outer face removed,
|
||||
and let $\mathcal{B}$ be the set of inner faces of $L_k$ whose bounding
|
||||
level cycle contains at least one edge of the outer cycle of $L_k$. The
|
||||
\emph{facial depth} of an inner face $F$ of $L_k$ is
|
||||
\[
|
||||
\mathrm{depth}(F) \;=\; \min_{F' \in \mathcal{B}} \mathrm{dist}_D(F, F'),
|
||||
\]
|
||||
with the convention $\mathrm{depth}(F) = \infty$ if no such $F'$ exists.
|
||||
An inner face is \emph{isolated} if $\mathrm{depth}(F) \geq 1$.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.6\textwidth]{fig_facial_depth.png}
|
||||
\caption{Facial depths in a maximal outerplanar graph on $12$ vertices.
|
||||
The six green ear-triangles share an edge with the outer $12$-cycle and
|
||||
so lie in $\mathcal{B}$ (depth $0$). The three yellow ``in-between''
|
||||
triangles $(0,2,4),(4,6,8),(0,8,10)$ have only diagonal edges but each
|
||||
is dual-adjacent to ears, giving them $\mathrm{depth} = 1$. The central
|
||||
triangle $(0,4,8)$ is also all-diagonal; its dual neighbours are the
|
||||
three depth-$1$ triangles, so it is isolated with
|
||||
$\mathrm{depth} = 2$.}
|
||||
\label{fig:facial-depth}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[Surface switch]
|
||||
\label{def:surface-switch}
|
||||
A \emph{surface switch} is an edge switch (Definition~\ref{def:edge-switch})
|
||||
applied to an edge incident to two level cycles, one of facial depth $d$
|
||||
and the other of facial depth $d - 1$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Balanced surface switch]
|
||||
\label{def:balanced-surface-switch}
|
||||
Let $\sigma$ be a surface switch on an edge $e = uv$ separating an inner
|
||||
face $F$ of $L_k$ of depth $d \geq 1$ from an adjacent inner face
|
||||
$F' = uvx$ of depth $d - 1$. We say $\sigma$ is \emph{balanced} if each
|
||||
of the two edges of $\partial F'$ other than $uv$ (namely $ux$ and $vx$)
|
||||
either lies on the outer cycle of $L_k$ or is shared with an inner face
|
||||
of $L_k$ of depth $d - 2$.
|
||||
\end{definition}
|
||||
|
||||
When $d = 1$ the condition reduces to ``both $ux$ and $vx$ lie on the
|
||||
outer cycle of $L_k$'', because no inner face has depth $-1$; in that
|
||||
case $F'$ is a triangular ``ear'' hanging off $uv$.
|
||||
|
||||
\section{Outerplanarity of level components}
|
||||
\label{sec:outerplanar-components}
|
||||
|
||||
For each integer $k \geq 0$ and each $(G, S)$, write $L_k$ for the
|
||||
subgraph of $G$ induced by the level-$k$ vertices. A \emph{level
|
||||
component} of $G$ (with respect to $S$) is a connected component of
|
||||
some $L_k$.
|
||||
|
||||
\begin{theorem}
|
||||
\label{thm:outerplanar-component}
|
||||
For every plane triangulation $G$ and every level source $S$ of $G$,
|
||||
every level component of $G$ is outerplanar.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
Since every subgraph of an outerplanar graph is outerplanar, it suffices
|
||||
to show that each level subgraph $L_k$ is outerplanar. For $k = 0$,
|
||||
$L_0$ is either a single vertex (when $S$ is a degree-$3$ vertex) or
|
||||
the triangle bounding the source face (when $S$ is a face), both
|
||||
outerplanar.
|
||||
|
||||
Fix $k \geq 1$ and let $D_k$ be the drawing of $L_k$ inherited from
|
||||
$\Pi_G$. Let $F^\ast$ be the face of $D_k$ containing the source.
|
||||
Suppose for contradiction that some $u \in L_k$ does not lie on
|
||||
$\partial F^\ast$, so $u$ lies on the boundary of some other face of
|
||||
$D_k$. Take any path $P$ in $G$ from $v_0 \in S$ to $u$. As a curve in
|
||||
$\Pi_G$, $P$ starts in $F^\ast$ and ends at a point off $\partial
|
||||
F^\ast$, so it must transition from $F^\ast$ to a different face of
|
||||
$D_k$; in a planar embedding this can happen only at a vertex of
|
||||
$D_k$, that is, at a level-$k$ vertex $w$ on $P$. Either $w \neq u$
|
||||
(so $P$ has length $\geq \mathrm{dist}_G(S, w) + 1 \geq k + 1$), or
|
||||
$w = u$ (contradicting $u \notin \partial F^\ast$). Since every
|
||||
$S$-to-$u$ path has length $\geq k + 1$, $\mathrm{dist}_G(S, u) \geq
|
||||
k + 1$, contradicting $u \in L_k$.
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}
|
||||
\label{lem:depth-descent}
|
||||
Let $C$ be a level component of $G$ with respect to $S$, drawn with the
|
||||
outerplanar embedding inherited from $\Pi_G$, and let $D$ be its
|
||||
inner-face dual. If $F$ is an inner face of $C$ with
|
||||
$\mathrm{depth}(F) = d > 0$, then $F$ is dual-adjacent to an inner
|
||||
face $F'$ with $\mathrm{depth}(F') = d - 1$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
By Theorem~\ref{thm:outerplanar-component}, $C$ is outerplanar, so the
|
||||
inner-face dual $D$ is a forest (a standard fact; a tree when $C$ is
|
||||
$2$-connected).
|
||||
|
||||
Each leaf $F_\ell$ of $D$ contains a single interior edge of $C$, so
|
||||
the remaining edges of $\partial F_\ell$ lie on the outer cycle of $C$.
|
||||
In particular $F_\ell$ has at least one outer-cycle edge, so
|
||||
$F_\ell \in \mathcal{B}$ and $\mathrm{depth}(F_\ell) = 0$. Hence every
|
||||
tree component of $D$ contains an element of $\mathcal{B}$, so the
|
||||
depths of all of its vertices are finite.
|
||||
|
||||
Choose a shortest path $F = F_0, F_1, \ldots, F_d = F^\ast$ in $D$ from
|
||||
$F$ to some $F^\ast \in \mathcal{B}$ realising $\mathrm{depth}(F) = d$.
|
||||
The suffix $F_1, \ldots, F_d$ witnesses $\mathrm{depth}(F_1) \leq d - 1$.
|
||||
If $\mathrm{depth}(F_1) \leq d - 2$, prepending the edge $F\,F_1$ to a
|
||||
witnessing path would give $\mathrm{depth}(F) \leq d - 1$, contradicting
|
||||
$\mathrm{depth}(F) = d$. Hence $\mathrm{depth}(F_1) = d - 1$, and we
|
||||
may take $F' := F_1$.
|
||||
\end{proof}
|
||||
|
||||
\begin{proposition}
|
||||
\label{prop:balanced-descent}
|
||||
Let $\sigma$ be a balanced surface switch on the edge $e = uv$ separating
|
||||
$F$ (depth $d \geq 1$) from $F' = uvx$ (depth $d - 1$), and let
|
||||
$G' = G - uv + wx$ be the result of the underlying edge flip, with
|
||||
$uvw, uvx$ the two triangular faces of $G$ at $uv$. Then in $L_k'$ (the
|
||||
level-$k$ subgraph of $G'$, with the level assignment of $G$):
|
||||
\begin{enumerate}
|
||||
\item the level cycle $\partial F$ is destroyed; and
|
||||
\item one or two new inner faces appear in $L_k'$, each of depth exactly
|
||||
$d - 1$.
|
||||
\end{enumerate}
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
The flip removes $uv$ from $G$, so $\partial F$ is no longer a cycle of
|
||||
$L_k'$, proving (1). For (2) we split on whether the new edge $wx$
|
||||
re-enters $L_k$.
|
||||
|
||||
\textbf{Case (i): $\{w, x\} \not\subseteq L_k$.} Then $L_k' = L_k - uv$.
|
||||
The faces $F$ and $F'$ merge into a single new inner face $\widetilde F$
|
||||
with boundary $(\partial F \cup \partial F') \setminus \{uv\}$. The dual
|
||||
neighbours of $\widetilde F$ in $L_k'$ are exactly the former neighbours
|
||||
of $F$ and $F'$ other than each other; in particular they include all
|
||||
inner faces previously adjacent to $F'$ across $ux$ or $vx$, whose depths
|
||||
are at most $d - 2$ by Lemma~\ref{lem:depth-descent} applied to $F'$
|
||||
(when $d \geq 2$). Thus $\mathrm{depth}(\widetilde F) \leq d - 1$.
|
||||
|
||||
For the matching lower bound, every neighbour of $\widetilde F$ has depth
|
||||
$\geq d - 2$ (neighbours inherited from $F$ have depth $\geq d - 1$;
|
||||
neighbours inherited from $F'$ have depth $\geq d - 2$). When $d \geq 2$,
|
||||
neither $F$ nor $F'$ has an outer-cycle edge, so neither does
|
||||
$\widetilde F$, giving $\mathrm{depth}(\widetilde F) \geq d - 1$. When
|
||||
$d = 1$, $F' \in \mathcal{B}$ and its outer-cycle edge (necessarily
|
||||
distinct from the interior edge $uv$) survives on $\partial \widetilde F$,
|
||||
so $\widetilde F \in \mathcal{B}'$ and
|
||||
$\mathrm{depth}(\widetilde F) = 0 = d - 1$. In either case
|
||||
$\mathrm{depth}(\widetilde F) = d - 1$, giving the unique new face
|
||||
required by~(2).
|
||||
|
||||
\textbf{Case (ii): $\{w, x\} \subseteq L_k$.} Then $F = uvw$ and
|
||||
$F' = uvx$ are triangular faces of $L_k$, and $L_k' = L_k - uv + wx$.
|
||||
The chord $wx$ splits the quadrilateral $\partial(F \cup F')$ into two
|
||||
triangular faces $A = uwx$ and $B = vwx$ of $L_k'$. We show
|
||||
$\mathrm{depth}(A) = \mathrm{depth}(B) = d - 1$.
|
||||
|
||||
By symmetry it suffices to handle $A$. The dual neighbours of $A$ in
|
||||
$L_k'$ are $A_{uw}$ (the inner face across $uw$, unchanged from $L_k$),
|
||||
$A_{ux}$ (the inner face across $ux$, unchanged), and $B$ (across the
|
||||
new edge $wx$). By balancedness of $\sigma$ applied to the edge $ux$:
|
||||
\begin{itemize}
|
||||
\item if $ux$ lies on the outer cycle of $L_k$, it remains on the outer
|
||||
cycle of $L_k'$, so $A \in \mathcal{B}'$ and $\mathrm{depth}(A) = 0$
|
||||
(which equals $d - 1$ because the balanced-with-outer-cycle case forces
|
||||
$d = 1$); or
|
||||
\item if $A_{ux}$ is an inner face, balancedness gives
|
||||
$\mathrm{depth}(A_{ux}) = d - 2$, so
|
||||
$\mathrm{depth}(A) \leq 1 + (d - 2) = d - 1$.
|
||||
\end{itemize}
|
||||
|
||||
For the lower bound in the second sub-case ($d \geq 2$): $A$'s edges are
|
||||
$uw$ (an edge of $F$, interior because $F$ has depth $d \geq 1$), $wx$
|
||||
(new, not on the outer cycle), and $ux$ (interior in this sub-case), so
|
||||
$A \notin \mathcal{B}'$. Moreover every neighbour of $A$ has depth $\geq
|
||||
d - 2$: $A_{uw}$ inherits depth $\geq d - 1$ from being a former
|
||||
neighbour of $F$, $A_{ux}$ has depth $d - 2$, and $B$ has depth $\geq
|
||||
d - 2$ by the same argument applied symmetrically. Therefore
|
||||
$\mathrm{depth}(A) \geq d - 1$, and combined with the upper bound,
|
||||
$\mathrm{depth}(A) = d - 1$.
|
||||
\end{proof}
|
||||
|
||||
\subsection*{When does a balanced surface switch exist?}
|
||||
|
||||
For a chord $uv$ of a maximal outerplanar graph, the \emph{span} of $uv$
|
||||
is the minimum, over the two arcs from $u$ to $v$ on the outer cycle,
|
||||
of the number of outer-cycle vertices strictly between them.
|
||||
|
||||
\begin{observation}
|
||||
\label{obs:span1-balanced-d1}
|
||||
For $d = 1$, an inner face $F$ admits a balanced surface switch on some
|
||||
edge iff at least one edge of $F$ has span $1$ in the outer cycle of
|
||||
$L_k$. The opposite triangle across that edge -- using the single
|
||||
outer-cycle vertex between its endpoints -- is then an ear of $F$ in
|
||||
$\mathcal{B}$, satisfying the $d = 1$ form of
|
||||
Definition~\ref{def:balanced-surface-switch}.
|
||||
\end{observation}
|
||||
|
||||
The smallest maximal-outerplanar configuration violating this is a
|
||||
$9$-vertex outer cycle triangulated so that the unique interior face
|
||||
$F = (0,3,6)$ has spans $(2,2,2)$ on its three edges
|
||||
(Figure~\ref{fig:no-balanced}). Each depth-$0$ neighbour of $F$ carries
|
||||
exactly one outer-cycle edge, not two, so none qualifies as an ear of
|
||||
$F$; no balanced surface switch is available.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.55\textwidth]{fig_no_balanced_switch.png}
|
||||
\caption{$9$-vertex maximal outerplanar $L_k$. $F = (0,3,6)$ has
|
||||
$\mathrm{depth} = 1$ and all three of its edges have span $2$, so none
|
||||
of $F$'s depth-$0$ neighbours is an ear. No balanced surface switch is
|
||||
available on $F$.}
|
||||
\label{fig:no-balanced}
|
||||
\end{figure}
|
||||
|
||||
\subsection*{Preprocessing toward balanced switches}
|
||||
|
||||
When $F$ has depth $d \geq 1$ but admits no balanced surface switch,
|
||||
perform a single (unbalanced) surface switch on any edge of $F$ shared
|
||||
with a depth-$(d-1)$ neighbour. By Proposition~\ref{prop:balanced-descent}
|
||||
the result is at least one new depth-$(d-1)$ face; in Case~(ii) it is
|
||||
accompanied by a new depth-$d$ face $A$ that replaces $F$ as the next
|
||||
candidate. The hope is that the resulting $A$ admits a balanced
|
||||
surface switch, or that iterating the preprocessing eventually exposes
|
||||
one.
|
||||
|
||||
\begin{example}
|
||||
\label{ex:preprocessing}
|
||||
On the $9$-vertex example, the (unbalanced) surface switch on edge
|
||||
$uv = 03$ -- with $F' = (0,2,3)$, third vertex $x = 2$, and $w = 6$ --
|
||||
flips $03 \mapsto 26$ in $G$ and produces $A = (0,2,6)$ at depth $1$.
|
||||
The new face has spans $(1, 3, 2)$ on its edges, and the ear
|
||||
$(0,1,2)$ across the span-$1$ edge $02$ is now a balanced
|
||||
surface-switch target on $A$ (Figure~\ref{fig:preprocessing}).
|
||||
\end{example}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{fig_preprocessing.png}
|
||||
\caption{One step of preprocessing on the $9$-vertex example. Left:
|
||||
$F = (0,3,6)$ has no edge of span $1$; the chosen surface-switch edge
|
||||
$uv = 03$ (red) is unbalanced. Right: after the switch $03 \mapsto 26$
|
||||
(green), the new depth-$1$ face $A = (0,2,6)$ has its edge $02$ (red)
|
||||
at span $1$, exposing the ear $(0,1,2)$ as a balanced surface-switch
|
||||
target.}
|
||||
\label{fig:preprocessing}
|
||||
\end{figure}
|
||||
|
||||
We do not have a general termination theorem. The natural candidate
|
||||
monovariant for $d = 1$ is the minimum span among edges of the current
|
||||
depth-$1$ face that are shared with a depth-$0$ neighbour; in
|
||||
Example~\ref{ex:preprocessing} this drops from $2$ to $1$ in a single
|
||||
step. Whether such a monovariant strictly decreases under every
|
||||
unbalanced surface switch -- and a corresponding statement for $d \geq
|
||||
2$, where balancedness depends on depth-$(d-2)$ structure rather than
|
||||
just spans -- remains open.
|
||||
|
||||
\begin{question}
|
||||
\label{q:preprocessing-terminates}
|
||||
Does iterated preprocessing reach a balanced surface switch in finitely
|
||||
many steps from every initial configuration? Equivalently, is there a
|
||||
monovariant on the inner-face structure of $L_k$ that strictly decreases
|
||||
at every unbalanced surface switch on a maximum-depth face?
|
||||
\end{question}
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user