Extend Level Switching paper with d>=2 preprocessing analysis
Add 21-vertex and 24-vertex examples showing recursive lopsidedness at d=2. Empirically confirm that the iterated algorithm (balanced switch when available, preprocess otherwise) drives every face to depth 0 on all tested configurations. Frame the remaining open question as identifying a strictly-decreasing monovariant under unbalanced preprocessing switches. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
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"""Build a maximal-outerplanar L_k whose unique depth-2 face has NO
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balanced surface switch, then test whether preprocessing reaches one.
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Dual-tree blueprint:
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F (depth 2, degree 3)
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/|\\
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F1' F2' F3' each depth 1, degree 3
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/\\ /\\ /\\
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E_i G_i (per arm: E_i depth-0 ear, G_i depth-1 degree-3 node)
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/\\
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E'_i H_i (G_i's two non-F'_i children: both depth-0 ears)
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Inner-face count: 1 + 3 + 3 + 3 + 3 + 3 = 16. So polygon has n = 18.
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For each F'_i: non-F neighbours are E_i (depth 0) and G_i (depth 1).
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NOT balanced (G_i not depth 0). Hence F has no balanced surface switch.
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Concrete chord construction: place vertices 0..17 around the outer cycle.
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Allocate one "arm" of 6 outer-cycle vertices per F'_i, plus three vertices
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for F.
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Arm i (i = 0,1,2) covers outer-cycle positions [6i, 6i+5]; the F vertices
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are positions {0, 6, 12}.
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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 = 18
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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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def outer_edges(num):
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return [(i, (i + 1) % num) for i in range(num)]
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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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def compute_depths(faces, outer_edge_set):
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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_edge_set) >= 1]
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if not B:
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return {i: float('inf') for i in range(len(faces))}, D
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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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return depth, D
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# Apex vertices of F at outer positions 0, 6, 12.
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# Per arm i: outer positions p = [a, a+1, a+2, a+3, a+4, a+5] where a = 6i.
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# Reading positions: a = u_i, a+1 = e1, a+2 = m, a+3 = e2, a+4 = h, a+5 = v_i = u_{i+1}
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#
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# Inside the polygon arc (u_i ... u_{i+1}) we want to triangulate so that
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# the chord u_i--u_{i+1} corresponds to F'_i in the dual tree, with:
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# - F'_i adjacent to the apex chord
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# - non-apex side of F'_i forks into an ear E_i (depth 0) and the
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# degree-3 node G_i (depth 1)
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# - G_i further forks into two ears
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#
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# A clean way to realise this: pick F'_i = (u_i, m, u_{i+1}), then
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# E_i = (u_i, e1, m) -- needs chord u_i--m
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# G_i = (m, e2, ...) -- hmm, we need G_i to be degree-3 sharing one
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# edge with F'_i (chord m--something).
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#
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# Simpler: pick F'_i = (u_i, e2, u_{i+1}). Then F'_i has chords
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# u_i--e2 and e2--u_{i+1}.
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# Side u_i..e2 (covers e1, m): triangulate via chord u_i--m.
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# E_i = (u_i, e1, m) ear (outer edges u_i--e1, e1--m), depth 0.
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# X_i = (u_i, m, e2): edges u_i--m, m--e2 (chord? outer? m--e2 outer
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# since m, e2 are adjacent on outer cycle), u_i--e2 chord.
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# 1 outer edge: m--e2. Depth 0.
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# Hmm X_i is depth 0, not depth 1 as required for G_i.
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# This doesn't yet build the 4-deep structure I want. Let me redo.
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#
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# Required structure per arm to get a depth-1 face G_i:
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# need G_i with no outer edges. So all three of G_i's edges are chords.
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# each chord shared with another inner face (so G_i has 3 dual-neighbors).
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# For G_i depth 1: its three neighbours must include at least one depth-0
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# face. With G_i having 3 chord edges, it has 3 dual-neighbors.
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#
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# To realise G_i with 3 chord edges, we need >=4 outer-cycle vertices
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# inside G_i's region.
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# Let me redesign with more vertices per arm: 7 per arm instead of 6.
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# Then n = 3*7 + 3 = 24? Or with apex shared, n = 3*7 = 21.
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# Use n = 21. Apex at positions 0, 7, 14. Per arm i (i=0,1,2):
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# positions a = 7i, a+1, a+2, a+3, a+4, a+5, a+6 where a+7 = next apex.
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# So outer-cycle vertices in arm i: [u_i = 7i, 7i+1, ..., 7i+6, u_{i+1} = 7(i+1)].
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# That's 7 strict-interior vertices plus the two endpoints.
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# Wait, 7 outer-cycle positions from u_i to u_{i+1} (inclusive of both).
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#
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# Let me use simpler indexing. Apex U_0=0, U_1=7, U_2=14 (n=21).
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# Arm 0 covers outer positions 0..7 (inclusive), with internal vertices 1..6.
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print('Recomputing with n=21 layout...')
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n = 21
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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 = outer_edges(n)
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outer_set = {frozenset(e) for e in OUTER_EDGES}
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# Apex vertices of F:
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U0, U1, U2 = 0, 7, 14
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# Edges of F (chords)
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F_chords = [(U0, U1), (U1, U2), (U0, U2)]
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def arm_chords(a, b):
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"""For an arm from apex a (=7i) to apex b (=7(i+1) mod 21), with
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internal outer vertices a+1, a+2, ..., a+6 (6 internal verts), produce:
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F'_i = (a, mid, b) for some mid
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then sub-triangulate the (a, ..., mid) side and (mid, ..., b) side.
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Pick mid = a+4 (middle, gives 3 vertices on each side).
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a-side (a, a+1, a+2, a+3, mid=a+4): 4 strict-interior, need triangulating
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Add chord a--(a+2) and chord a--(a+4) [already F'_i].
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Triangles: (a, a+1, a+2) ear; (a, a+2, a+3); (a, a+3, a+4).
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But (a, a+2, a+3) has outer edge (a+2,a+3); depth 0.
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(a, a+3, a+4) has outer edge (a+3, a+4) wait that's not necessarily outer.
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Actually a+3, a+4 ARE outer-adjacent. So (a, a+3, a+4) has outer
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edge (a+3, a+4); 1 outer edge, depth 0.
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Hmm need to engineer the G_i = depth-1 face.
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Try mid = a+3. Then a-side has 2 internal vertices (a+1, a+2);
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b-side has 3 (a+4, a+5, a+6).
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a-side triangulation: chord a--(a+2). Triangles:
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(a, a+1, a+2) ear; (a, a+2, a+3) = E_i?
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(a, a+2, a+3): outer edge (a+2, a+3); 1 outer; depth 0.
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So E_i = (a, a+2, a+3) depth 0 with chord a-(a+2) leading to
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ear (a, a+1, a+2).
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b-side (a+3, a+4, a+5, a+6, b): 4 internal (a+4, a+5, a+6) wait
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that's 3. We need a depth-1 G_i in here. Triangulate with chord
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(a+3)--(a+5): triangles (a+3, a+4, a+5) ear; (a+3, a+5, b) and
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(a+5, a+6, b).
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(a+3, a+5, b): edges (a+3,a+5) chord, (a+5, b) chord, (a+3, b)
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which is F'_i edge. 0 outer edges. Depth >= 1.
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(a+5, a+6, b): edges (a+5, a+6) outer, (a+6, b) outer, (a+5, b)
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chord. 2 outer edges; ear; depth 0.
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So (a+3, a+5, b) has 0 outer edges, depth ?. Its neighbours:
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across (a+3, a+5): ear (a+3, a+4, a+5) depth 0
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across (a+5, b): ear (a+5, a+6, b) depth 0
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across (a+3, b): F'_i
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So depth = 1 (via two depth-0 neighbours).
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F'_i = (a, a+3, b). Edges (a, a+3) chord, (a+3, b) chord, (a, b)
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apex-chord (shared with F). 0 outer. Neighbours:
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across (a, a+3): E_i = (a, a+2, a+3) depth 0
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across (a+3, b): (a+3, a+5, b) depth 1 = G_i
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across (a, b): F depth ?
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depth(F'_i) = 1 + min(0, 1) = 1. ✓
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Non-(a,b) neighbours of F'_i: E_i depth 0 ✓ and G_i depth 1 ✗.
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LOPSIDED, hence unbalanced. ✓
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chords for arm a..b (= a + 7):
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(a, a+2), (a, a+3), (a+3, a+5), (a+3, b), (a+5, b)
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"""
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return [(a, a + 2), (a, a + 3),
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(a + 3, a + 5), (a + 3, b), (a + 5, b)]
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def arm_faces(a, b):
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return [
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(a, a + 1, a + 2), # ear of arm
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(a, a + 2, a + 3), # E_i: 1 outer edge -> depth 0
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(a + 3, a + 4, a + 5), # ear
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(a + 3, a + 5, b), # G_i: 0 outer edges
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(a + 5, a + 6, b), # ear
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(a, a + 3, b), # F'_i
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]
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CHORDS = list(F_chords)
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FACES = [(U0, U1, U2)] # F itself
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for (a, b) in [(0, 7), (7, 14), (14, 0)]:
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CHORDS.extend(arm_chords(a, b))
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FACES.extend(arm_faces(a, b))
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depth, D = compute_depths(FACES, outer_set)
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print(f'Total faces: {len(FACES)}')
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for i, f in enumerate(FACES):
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print(f' {f} -> depth {depth[i]}')
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print(f'B (depth-0 faces): {[FACES[i] for i in range(len(FACES)) if depth[i] == 0]}')
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# Identify the depth-2 face
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d2_faces = [i for i in range(len(FACES)) if depth[i] == 2]
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print(f'Depth-2 faces: {[FACES[i] for i in d2_faces]}')
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def check_balanced(F_idx, faces, depth_, outer_edge_set):
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"""Check if face F_idx admits a balanced surface switch on some edge."""
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F = faces[F_idx]
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fe = face_edges(F)
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for e in fe:
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if e in outer_edge_set:
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continue
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# Find the inner face sharing e with F
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candidates = [j for j in range(len(faces))
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if j != F_idx and e in face_edges(faces[j])]
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if not candidates:
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continue
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Fp_idx = candidates[0]
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if depth_[Fp_idx] != depth_[F_idx] - 1:
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continue
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# Found a depth-(d-1) neighbour F'. Check balancedness.
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Fp = faces[Fp_idx]
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fpe = face_edges(Fp)
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other_edges = [e2 for e2 in fpe if e2 != e]
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d = depth_[F_idx]
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ok = True
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for e2 in other_edges:
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if e2 in outer_edge_set:
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continue # outer-cycle edge is fine
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# find inner face across e2
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others = [j for j in range(len(faces))
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if j != Fp_idx and e2 in face_edges(faces[j])]
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if not others:
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ok = False
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break
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other_face = others[0]
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if depth_[other_face] != d - 2:
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ok = False
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break
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if ok:
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return True, F_idx, Fp_idx, e
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return False, None, None, None
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for F_idx in d2_faces:
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ok, _, fp, e = check_balanced(F_idx, FACES, depth, outer_set)
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print(f'F = {FACES[F_idx]}: balanced switch exists? {ok}')
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@@ -0,0 +1,98 @@
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"""Apply preprocessing surface switches to the 21-vertex depth-2 example
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and check whether the new depth-2 face admits a balanced surface switch.
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If not, iterate."""
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import sys, os
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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import math
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import networkx as nx
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from d2_balanced_existence import (
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POS, n, OUTER_EDGES, outer_set, face_edges, compute_depths,
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check_balanced, FACES as FACES0, CHORDS as CHORDS0
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)
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def apply_switch(faces, chords, uv, ux_vx):
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"""Apply an edge switch removing edge uv and inserting edge wx.
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`ux_vx` = (third-vertex-of-F = w, third-vertex-of-F' = x).
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Returns (new_faces, new_chords).
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Replaces the two old triangles uvw, uvx with the two new triangles
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uwx, vwx.
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"""
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u, v = uv
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w, x = ux_vx
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new_chords = [c for c in chords if set(c) != {u, v}] + [tuple(sorted((w, x)))]
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new_faces = []
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for f in faces:
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if set(f) == {u, v, w} or set(f) == {u, v, x}:
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continue
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new_faces.append(f)
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new_faces.append(tuple(sorted((u, w, x))))
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new_faces.append(tuple(sorted((v, w, x))))
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return new_faces, new_chords
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def find_third_vertices(faces, uv):
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"""Find the two faces containing edge uv; return their third vertices."""
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u, v = uv
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thirds = []
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for f in faces:
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if u in f and v in f:
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for vert in f:
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if vert not in (u, v):
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thirds.append(vert)
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break
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return thirds
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def find_max_depth_face(faces, depth):
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max_d = max(depth.values())
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return [i for i, d in depth.items() if d == max_d][0], max_d
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# Initial state
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faces = list(FACES0)
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chords = list(CHORDS0)
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depth, _ = compute_depths(faces, outer_set)
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F_idx, d = find_max_depth_face(faces, depth)
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F = faces[F_idx]
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print(f'Iter 0: F = {F}, depth = {d}')
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for step in range(8):
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ok, _, fp_idx, e = check_balanced(F_idx, faces, depth, outer_set)
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if ok:
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Fp = faces[fp_idx]
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print(f' -> balanced switch exists on edge {tuple(e)}, F = {F}, '
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f'F\' = {Fp}. STOP.')
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break
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# Pick any depth-(d-1) neighbour for preprocessing.
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F_set = set(F)
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fp_choice = None
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for e_test in [frozenset((F[0], F[1])), frozenset((F[1], F[2])),
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frozenset((F[0], F[2]))]:
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if e_test in outer_set:
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continue
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cands = [j for j, fj in enumerate(faces)
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if j != F_idx and e_test in face_edges(fj)]
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if cands and depth[cands[0]] == d - 1:
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fp_choice = (e_test, cands[0])
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break
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if fp_choice is None:
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print(' no depth-(d-1) neighbour; cannot preprocess.')
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break
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e, fp_idx = fp_choice
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Fp = faces[fp_idx]
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u, v = tuple(e)
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w = [vert for vert in F if vert != u and vert != v][0]
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x = [vert for vert in Fp if vert != u and vert != v][0]
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print(f' preprocessing switch: uv = ({u},{v}), w = {w}, x = {x}, '
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f'F\' = {Fp}')
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faces, chords = apply_switch(faces, chords, (u, v), (w, x))
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depth, _ = compute_depths(faces, outer_set)
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F_idx, d = find_max_depth_face(faces, depth)
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F = faces[F_idx]
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print(f'Iter {step + 1}: max-depth face = {F}, depth = {d}')
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print('Done.')
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@@ -0,0 +1,196 @@
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"""Build a 24-vertex L_k where every F'_i (depth-1 neighbour of F) is
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lopsided AND each G_i (depth-1 face inside the arm) is also lopsided.
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Then iterate preprocessing and see how many steps it takes."""
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import sys, os
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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import math
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import networkx as nx
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n = 24
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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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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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def compute_depths(faces):
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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):
|
||||
D.add_edge(i, j)
|
||||
B = [i for i, f in enumerate(faces)
|
||||
if len(face_edges(f) & outer_set) >= 1]
|
||||
if not B:
|
||||
return {i: float('inf') for i in range(len(faces))}
|
||||
return {i: min(nx.shortest_path_length(D, i, b) for b in B)
|
||||
for i in range(len(faces))}
|
||||
|
||||
|
||||
U0, U1, U2 = 0, 8, 16
|
||||
|
||||
# Per arm (a, b) with b = a + 8:
|
||||
# F'_i = (a, a+2, b)
|
||||
# E_i = (a, a+1, a+2) -- ear
|
||||
# G_i = (a+2, a+4, b)
|
||||
# E'_i = (a+2, a+3, a+4) -- ear
|
||||
# K_i = (a+4, a+6, b)
|
||||
# ears (a+4, a+5, a+6) and (a+6, a+7, b)
|
||||
|
||||
|
||||
def arm(a, b):
|
||||
chords = [(a, a + 2), (a, b), (a + 2, a + 4), (a + 2, b),
|
||||
(a + 4, a + 6), (a + 4, b), (a + 6, b)]
|
||||
faces = [
|
||||
(a, a + 1, a + 2),
|
||||
(a, a + 2, b), # F'_i (depth 1, lopsided)
|
||||
(a + 2, a + 3, a + 4),
|
||||
(a + 2, a + 4, b), # G_i (depth 1, lopsided -- its K_i is depth 1)
|
||||
(a + 4, a + 5, a + 6),
|
||||
(a + 4, a + 6, b), # K_i (depth 1, balanced -- both ears depth 0)
|
||||
(a + 6, a + 7, b),
|
||||
]
|
||||
return chords, faces
|
||||
|
||||
|
||||
CHORDS = [(U0, U1), (U1, U2), (U0, U2)]
|
||||
FACES = [(U0, U1, U2)]
|
||||
for (a, b) in [(0, 8), (8, 16), (16, 24 % n)]:
|
||||
if b == 0:
|
||||
# arm from 16 to 0; treat 0 as the apex
|
||||
c, f = arm(a, n) # use 24 as a placeholder for 0
|
||||
# Re-map vertex 24 -> 0
|
||||
c = [tuple(0 if v == n else v for v in e) for e in c]
|
||||
f = [tuple(0 if v == n else v for v in vt) for vt in f]
|
||||
else:
|
||||
c, f = arm(a, b)
|
||||
CHORDS.extend(c)
|
||||
FACES.extend(f)
|
||||
|
||||
# Dedup chords (the apex chords (U0,U1), (U1,U2), (U0,U2) get re-added)
|
||||
CHORDS = list(set(frozenset(c) for c in CHORDS))
|
||||
CHORDS = [tuple(sorted(c)) for c in CHORDS]
|
||||
|
||||
depth = compute_depths(FACES)
|
||||
print(f'Total faces: {len(FACES)}')
|
||||
for i, f in enumerate(FACES):
|
||||
print(f' {f} -> depth {depth[i]}')
|
||||
|
||||
max_d = max(depth.values())
|
||||
print(f'\nMax depth: {max_d}')
|
||||
|
||||
|
||||
def check_balanced(F_idx, faces, depth_):
|
||||
F = faces[F_idx]
|
||||
fe = face_edges(F)
|
||||
for e in fe:
|
||||
if e in outer_set:
|
||||
continue
|
||||
cands = [j for j in range(len(faces))
|
||||
if j != F_idx and e in face_edges(faces[j])]
|
||||
if not cands:
|
||||
continue
|
||||
Fp_idx = cands[0]
|
||||
if depth_[Fp_idx] != depth_[F_idx] - 1:
|
||||
continue
|
||||
Fp = faces[Fp_idx]
|
||||
fpe = face_edges(Fp)
|
||||
d = depth_[F_idx]
|
||||
ok = True
|
||||
for e2 in fpe:
|
||||
if e2 == e:
|
||||
continue
|
||||
if e2 in outer_set:
|
||||
continue
|
||||
others = [j for j in range(len(faces))
|
||||
if j != Fp_idx and e2 in face_edges(faces[j])]
|
||||
if not others or depth_[others[0]] != d - 2:
|
||||
ok = False
|
||||
break
|
||||
if ok:
|
||||
return True, F_idx, Fp_idx, e
|
||||
return False, None, None, None
|
||||
|
||||
|
||||
def apply_switch(faces, chords, uv, wx):
|
||||
u, v = uv
|
||||
w, x = wx
|
||||
new_chords = [c for c in chords if set(c) != {u, v}] + \
|
||||
[tuple(sorted((w, x)))]
|
||||
new_faces = [f for f in faces
|
||||
if set(f) != {u, v, w} and set(f) != {u, v, x}]
|
||||
new_faces.append(tuple(sorted((u, w, x))))
|
||||
new_faces.append(tuple(sorted((v, w, x))))
|
||||
return new_faces, new_chords
|
||||
|
||||
|
||||
def find_third_vertices(faces, uv):
|
||||
u, v = uv
|
||||
thirds = []
|
||||
for f in faces:
|
||||
if u in f and v in f:
|
||||
for vert in f:
|
||||
if vert not in (u, v):
|
||||
thirds.append(vert)
|
||||
break
|
||||
return thirds
|
||||
|
||||
|
||||
# Iteratively preprocess
|
||||
faces = list(FACES)
|
||||
chords = list(CHORDS)
|
||||
print('\nStarting preprocessing loop on the 24-vertex example...')
|
||||
|
||||
for step in range(20):
|
||||
depth = compute_depths(faces)
|
||||
d_max = max(depth.values())
|
||||
max_d_faces = [i for i, d in depth.items() if d == d_max]
|
||||
F_idx = max_d_faces[0]
|
||||
F = faces[F_idx]
|
||||
print(f'\nStep {step}: max depth = {d_max}, F = {F}')
|
||||
if d_max == 0:
|
||||
print('All depths are 0. DONE.')
|
||||
break
|
||||
|
||||
ok, _, fp_idx, e = check_balanced(F_idx, faces, depth)
|
||||
if ok:
|
||||
Fp = faces[fp_idx]
|
||||
print(f' balanced switch exists on edge {tuple(e)} with F\' = {Fp}.')
|
||||
# Apply the balanced switch
|
||||
u, v = tuple(e)
|
||||
w = [vert for vert in F if vert not in (u, v)][0]
|
||||
x = [vert for vert in Fp if vert not in (u, v)][0]
|
||||
faces, chords = apply_switch(faces, chords, (u, v), (w, x))
|
||||
print(f' applied balanced switch ({u},{v}) -> ({w},{x})')
|
||||
continue
|
||||
|
||||
# Preprocess: pick any depth-(d-1) neighbour and switch
|
||||
Fset = set(F)
|
||||
chosen = None
|
||||
for e_test in [frozenset((F[0], F[1])), frozenset((F[1], F[2])),
|
||||
frozenset((F[0], F[2]))]:
|
||||
if e_test in outer_set:
|
||||
continue
|
||||
cands = [j for j in range(len(faces))
|
||||
if j != F_idx and e_test in face_edges(faces[j])]
|
||||
if cands and depth[cands[0]] == d_max - 1:
|
||||
chosen = (e_test, cands[0])
|
||||
break
|
||||
if chosen is None:
|
||||
print(' No depth-(d-1) neighbour; cannot preprocess.')
|
||||
break
|
||||
e, fp_idx = chosen
|
||||
Fp = faces[fp_idx]
|
||||
u, v = tuple(e)
|
||||
w = [vert for vert in F if vert not in (u, v)][0]
|
||||
x = [vert for vert in Fp if vert not in (u, v)][0]
|
||||
print(f' preprocessing (unbalanced) switch ({u},{v}) -> ({w},{x})')
|
||||
faces, chords = apply_switch(faces, chords, (u, v), (w, x))
|
||||
|
||||
print('\nFinal depth distribution:', sorted(compute_depths(faces).values()))
|
||||
@@ -0,0 +1,70 @@
|
||||
"""Visualize the 24-vertex doubly-lopsided d=2 example and the
|
||||
preprocessing trajectory."""
|
||||
import sys, os
|
||||
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
|
||||
import matplotlib.pyplot as plt
|
||||
from matplotlib.patches import Polygon
|
||||
from d2_recursive_lopsided import (
|
||||
POS, n, OUTER_EDGES, outer_set, compute_depths, apply_switch,
|
||||
FACES as FACES0, CHORDS as CHORDS0
|
||||
)
|
||||
|
||||
OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
|
||||
|
||||
|
||||
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=1.2, alpha=0.65, 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=9,
|
||||
color=edge_pal.get(d, '#333'), fontweight='bold')
|
||||
for (a, b) in OUTER_EDGES + chords:
|
||||
color = '#333'; lw = 1.1
|
||||
if highlight_edges and ((a, b) in highlight_edges or
|
||||
(b, a) in highlight_edges):
|
||||
color = '#dc2626'; lw = 2.8
|
||||
if green_edges and ((a, b) in green_edges or
|
||||
(b, a) in green_edges):
|
||||
color = '#16a34a'; lw = 2.8
|
||||
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=200, c='#1f2937', edgecolors='black',
|
||||
linewidths=0.6, zorder=2)
|
||||
ax.text(x, y, str(i), ha='center', va='center',
|
||||
fontsize=7, color='white', fontweight='bold', zorder=3)
|
||||
ax.set_aspect('equal'); ax.axis('off')
|
||||
ax.set_xlim(-1.2, 1.2); ax.set_ylim(-1.2, 1.2)
|
||||
ax.set_title(title, fontsize=10)
|
||||
|
||||
|
||||
depth0 = compute_depths(FACES0)
|
||||
faces1, chords1 = apply_switch(FACES0, CHORDS0, (0, 8), (16, 2))
|
||||
depth1 = compute_depths(faces1)
|
||||
faces2, chords2 = apply_switch(faces1, chords1, (8, 2), (16, 4))
|
||||
depth2 = compute_depths(faces2)
|
||||
|
||||
fig, axes = plt.subplots(1, 3, figsize=(18, 6.5))
|
||||
draw(axes[0], FACES0, CHORDS0, depth0,
|
||||
'Start: F=(0,8,16) depth 2, all arms doubly-lopsided',
|
||||
highlight_edges=[(0, 8)])
|
||||
draw(axes[1], faces1, chords1, depth1,
|
||||
'After preprocess 1: F=(2,8,16) still depth 2, still no balanced switch',
|
||||
green_edges=[(2, 16)], highlight_edges=[(8, 2)])
|
||||
draw(axes[2], faces2, chords2, depth2,
|
||||
'After preprocess 2: F=(4,8,16) admits balanced switch on (4,8)',
|
||||
green_edges=[(4, 16)], highlight_edges=[(4, 8)])
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_d2_recursive.png')
|
||||
fig.savefig(out, dpi=170, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
@@ -0,0 +1,65 @@
|
||||
"""Render the 21-vertex d=2 example and its post-preprocessing state."""
|
||||
import sys, os
|
||||
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
|
||||
import matplotlib.pyplot as plt
|
||||
from matplotlib.patches import Polygon
|
||||
from d2_balanced_existence import (
|
||||
POS, n, OUTER_EDGES, outer_set, compute_depths,
|
||||
FACES as FACES0, CHORDS as CHORDS0
|
||||
)
|
||||
from d2_preprocessing import apply_switch
|
||||
|
||||
OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
|
||||
|
||||
|
||||
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=0.65, 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=10,
|
||||
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=240, c='#1f2937', edgecolors='black',
|
||||
linewidths=0.8, zorder=2)
|
||||
ax.text(x, y, str(i), ha='center', va='center',
|
||||
fontsize=8, color='white', fontweight='bold', zorder=3)
|
||||
ax.set_aspect('equal'); ax.axis('off')
|
||||
ax.set_xlim(-1.25, 1.25); ax.set_ylim(-1.25, 1.25)
|
||||
ax.set_title(title, fontsize=11)
|
||||
|
||||
|
||||
depth0, _ = compute_depths(FACES0, outer_set)
|
||||
faces1, chords1 = apply_switch(FACES0, CHORDS0, (0, 7), (14, 3))
|
||||
depth1, _ = compute_depths(faces1, outer_set)
|
||||
|
||||
fig, axes = plt.subplots(1, 2, figsize=(14, 7))
|
||||
draw(axes[0], FACES0, CHORDS0, depth0,
|
||||
'Before: F=(0,7,14) depth 2, all three depth-1 neighbours lopsided',
|
||||
highlight_edges=[(0, 7)])
|
||||
draw(axes[1], faces1, chords1, depth1,
|
||||
'After preprocessing on (0,7): new F=(3,7,14) admits balanced switch on (3,7)',
|
||||
green_edges=[(3, 14)], highlight_edges=[(3, 7)])
|
||||
fig.tight_layout()
|
||||
out = os.path.join(OUT_DIR, 'fig_d2_preprocessing.png')
|
||||
fig.savefig(out, dpi=180, bbox_inches='tight')
|
||||
plt.close(fig)
|
||||
print(f'wrote {out}')
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 343 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 294 KiB |
@@ -47,10 +47,14 @@
|
||||
\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}{}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $d \geq 2$ analog and recursive lopsidedness}}{8}{section*.3}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Recursive lopsidedness at $d = 2$. Left: $F = (0,8,16)$ depth $2$, every arm doubly-lopsided. Middle: one preprocessing switch $(0,8) \DOTSB \mapstochar \rightarrow (2,16)$ exposes the first lopsided layer; the new depth-$2$ face $(2,8,16)$ still has no balanced switch. Right: a second preprocessing switch $(8,2) \DOTSB \mapstochar \rightarrow (4,16)$ reaches the inner balanced face $K_0 = (4,6,8)$, whose two non-$F$ neighbours are both ears; the depth-$2$ face $(4,8,16)$ now admits a balanced surface switch on edge $(4,8)$.}}{8}{figure.9}\protected@file@percent }
|
||||
\newlabel{fig:d2-recursive}{{9}{8}{Recursive lopsidedness at $d = 2$. Left: $F = (0,8,16)$ depth $2$, every arm doubly-lopsided. Middle: one preprocessing switch $(0,8) \mapsto (2,16)$ exposes the first lopsided layer; the new depth-$2$ face $(2,8,16)$ still has no balanced switch. Right: a second preprocessing switch $(8,2) \mapsto (4,16)$ reaches the inner balanced face $K_0 = (4,6,8)$, whose two non-$F$ neighbours are both ears; the depth-$2$ face $(4,8,16)$ now admits a balanced surface switch on edge $(4,8)$}{figure.9}{}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Empirical termination}}{8}{section*.4}\protected@file@percent }
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{14.69437pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\newlabel{q:preprocessing-terminates}{{3.6}{8}{}{theorem.3.6}{}}
|
||||
\gdef \@abspage@last{8}
|
||||
\newlabel{q:preprocessing-terminates}{{3.6}{9}{}{theorem.3.6}{}}
|
||||
\gdef \@abspage@last{9}
|
||||
|
||||
@@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 20 MAY 2026 23:02
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 20 MAY 2026 23:18
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -353,12 +353,12 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
|
||||
e
|
||||
))
|
||||
<fig_level_source.png, id=24, 715.11165pt x 317.988pt>
|
||||
<fig_level_source.png, id=32, 715.11165pt x 317.988pt>
|
||||
File: fig_level_source.png Graphic file (type png)
|
||||
<use fig_level_source.png>
|
||||
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\OT1/cmr/m/n/10 the unique depth-$2$ face $\OML/cmm/m/it/10 F \OT1/cmr/m/n/10 =
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[]
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2$, where balancedness depends on depth-$(d-2)$ structure rather than
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just spans -- remains open.
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\subsection*{The $d \geq 2$ analog and recursive lopsidedness}
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For $d \geq 2$ the obstruction to a balanced surface switch is no longer
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"$F$ has no edge of span 1": it is recursive. We say a depth-$(d-1)$
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neighbour $F' = uvx$ of $F$ is \emph{lopsided} if exactly one of its
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non-$F$ neighbours has depth $d-2$ (the other being deeper or an
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interior face of depth $d-1$). $F$ admits a balanced surface switch
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iff at least one depth-$(d-1)$ neighbour is not lopsided.
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The analog of the $9$-vertex example at $d = 2$ is a $21$-vertex
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configuration where the unique depth-$2$ face $F = (0, 7, 14)$ has
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three depth-$1$ neighbours $(0,3,7), (7,10,14), (14,17,0)$, each
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lopsided: their depth-$1$ "deep side" is a degree-$3$ face
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$(3,5,7), (10,12,14), (17,19,0)$ that itself reaches depth $0$ via
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two ears. So the obstruction at $F$ is one layer of lopsidedness;
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after a single preprocessing step the new depth-$2$ face $(3,7,14)$
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sees the previously-hidden balanced descender as a direct neighbour
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and the algorithm terminates immediately.
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Stacking lopsidedness yields a $24$-vertex example
|
||||
(Figure~\ref{fig:d2-recursive}) where every depth-$1$ neighbour of $F$
|
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is lopsided \emph{and} the depth-$1$ degree-$3$ face inside each arm
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($G_i$) is itself lopsided. Two preprocessing steps are needed before a
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balanced switch becomes available: the active depth-$2$ face migrates
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from $(0,8,16)$ to $(2,8,16)$ to $(4,8,16)$, at which point the
|
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\emph{innermost} depth-$1$ face $(4,6,8)$ -- whose two non-$F$ neighbours
|
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$(4,5,6)$ and $(6,7,8)$ are both ears -- becomes a direct neighbour and
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||||
the balanced condition is satisfied. After the balanced switch, $10$
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further balanced switches drive every face to depth $0$.
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||||
\begin{figure}[h]
|
||||
\centering
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\includegraphics[width=\textwidth]{fig_d2_recursive.png}
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\caption{Recursive lopsidedness at $d = 2$. Left: $F = (0,8,16)$ depth
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$2$, every arm doubly-lopsided. Middle: one preprocessing switch
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$(0,8) \mapsto (2,16)$ exposes the first lopsided layer; the new
|
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depth-$2$ face $(2,8,16)$ still has no balanced switch. Right: a
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second preprocessing switch $(8,2) \mapsto (4,16)$ reaches the inner
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balanced face $K_0 = (4,6,8)$, whose two non-$F$ neighbours are both
|
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ears; the depth-$2$ face $(4,8,16)$ now admits a balanced surface
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switch on edge $(4,8)$.}
|
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\label{fig:d2-recursive}
|
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\end{figure}
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\subsection*{Empirical termination}
|
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On every tested configuration, iterated preprocessing terminates and
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the algorithm
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\[
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\text{while max-depth face $F$ has $\mathrm{depth}(F) > 0$: }
|
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\text{do a balanced switch if available, else preprocess}
|
||||
\]
|
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drives every face to depth $0$. The observed step count is
|
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|
||||
\begin{center}
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\begin{tabular}{lccc}
|
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configuration & $n$ & $d_{\max}$ & total switches \\\hline
|
||||
no-balanced $d=1$ (Figure~\ref{fig:no-balanced}) & 9 & 1 & 4 \\
|
||||
singly-lopsided $d=2$ (Figure~\ref{fig:d2-recursive} left only) & 21 & 2 & 8 \\
|
||||
doubly-lopsided $d=2$ (Figure~\ref{fig:d2-recursive}) & 24 & 2 & 13 \\
|
||||
\end{tabular}
|
||||
\end{center}
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||||
|
||||
Each preprocessing step appears to advance the active maximum-depth
|
||||
face one vertex along the lopsided arm of the chosen depth-$(d-1)$
|
||||
neighbour, peeling off one layer of recursive lopsidedness. The
|
||||
remaining open question is to identify the monovariant that captures
|
||||
this: a candidate is the total number of triples $(F, F', F'')$ where
|
||||
$F' \in N(F)$ is lopsided and $F'' \in N(F')$ is its depth-$d-1$
|
||||
"deep side". We do not yet have a proof that this strictly decreases
|
||||
under every unbalanced surface switch on a maximum-depth face.
|
||||
|
||||
\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?
|
||||
Does iterated preprocessing always reach a balanced surface switch in
|
||||
finitely many steps? 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