From 78afa9aeeb0c5a667b91d4a565aac09f1e81c2f7 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 16:37:48 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20drop=20(R1?= =?UTF-8?q?)=20=E2=80=94=20empirically=20all=20"pinches"=20are=20cut-verti?= =?UTF-8?q?ces=20of=20O,=20already=20accommodated?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Empirical evidence (from check_r2.py + classify_r1_pinches.py over maximal planar graphs in n ∈ [7, 11]): Total components: 47,253 Total claimed (R1) pinches: 1,319 All 1,319 pinches are at level-(d+1) vertices that are cut-vertices of O = G[V_{C'} ∩ L_{d+1}]. ZERO level-d pinches (which would have been a problem for B_out). Under the relaxed Definition 1.5 (where B_in is the outer-face boundary closed walk of O, not necessarily a simple cycle), cut- vertices of O are naturally accommodated: the closed walk visits the cut-vertex multiple times. So what I previously called "(R1) violations" are not obstructions at all — they're just structural features of O that the closed-walk B_in captures. Changes: - Lemma 1.7: dropped (R1) hypothesis. Lemma is now unconditional (modulo the BFS-on-the-outer-face embedding choice already in the setup). - Proof: rewritten boundary-structure paragraph to describe the cut-vertex case naturally instead of citing (R1). - Definition 1.5: removed the "2-manifold" assertion (since R is not a manifold at cut-vertices of O); added an explicit note that R may fail to be a 2-manifold at cut-vertices and that the closed walk B_in visits them multiple times. - Remark 1.9 (was rem:R1-when): rewritten as "no extra hypotheses needed", documenting that both cut-vertex / multi-arc structures and multi-hole topology are already accommodated by Definition 1.5. Adds experiments/check_r1_concrete.py and experiments/classify_r1_pinches.py for verification, plus experiments/draw_r1_candidate.py and r1_candidate.png showing the n=9 bowtie example concretely. Co-Authored-By: Claude Opus 4.7 --- .../experiments/check_r1_concrete.py | 122 +++++++++++++ .../experiments/classify_r1_pinches.py | 146 +++++++++++++++ .../experiments/draw_r1_candidate.py | 172 ++++++++++++++++++ .../experiments/r1_candidate.png | Bin 0 -> 111391 bytes papers/coloring_nested_tire_graphs/paper.aux | 4 +- papers/coloring_nested_tire_graphs/paper.log | 10 +- papers/coloring_nested_tire_graphs/paper.pdf | Bin 486084 -> 483570 bytes papers/coloring_nested_tire_graphs/paper.tex | 89 ++++----- 8 files changed, 486 insertions(+), 57 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/check_r1_concrete.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/classify_r1_pinches.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_r1_candidate.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/r1_candidate.png diff --git a/papers/coloring_nested_tire_graphs/experiments/check_r1_concrete.py b/papers/coloring_nested_tire_graphs/experiments/check_r1_concrete.py new file mode 100644 index 0000000..6f41060 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/check_r1_concrete.py @@ -0,0 +1,122 @@ +"""Find a concrete claimed (R1) violation and verify it by hand. + +Iterate over small maximal planar graphs, for each single-vertex source +find a depth-d component flagged as having a pinch vertex by the cyclic- +rotation check. Save the graph + the pinch vertex's rotation, the +depth of each face incident to the pinch vertex, and the cyclic +sequence of "is this face in the component?" booleans. +""" +import sys +from collections import defaultdict + +from sage.all import Graph +from sage.graphs.graph_generators import graphs + + +def face_edges_set(face): + return frozenset(frozenset(e) for e in face) + + +def main(): + for n in range(7, 12): + for G in graphs.triangulations(n): + G.is_planar(set_embedding=True) + emb = G.get_embedding() + faces = G.faces(embedding=emb) + for v_source in G.vertices(): + dist = G.shortest_path_lengths(v_source) + face_depths = [] + for face in faces: + verts = set() + for u, v in face: + verts.add(u); verts.add(v) + d = min(dist[u] for u in verts) + face_depths.append((face, d, verts)) + max_d = max(fd[1] for fd in face_depths) + for target_d in range(max_d + 1): + cands = [(i, fd) for i, fd in enumerate(face_depths) + if fd[1] == target_d] + if not cands: + continue + edge_to_idxs = defaultdict(list) + for k, (i, fd) in enumerate(cands): + for e in [frozenset(ed) for ed in fd[0]]: + edge_to_idxs[e].append(k) + n_c = len(cands) + adj = [[] for _ in range(n_c)] + for idxs in edge_to_idxs.values(): + if len(idxs) >= 2: + for a in idxs: + for b in idxs: + if a != b: + adj[a].append(b) + visited = [False] * n_c + comps = [] + for s in range(n_c): + if visited[s]: + continue + stack = [s]; comp = [] + while stack: + x = stack.pop() + if visited[x]: continue + visited[x] = True; comp.append(cands[x]) + for y in adj[x]: + if not visited[y]: stack.append(y) + comps.append(comp) + + for comp in comps: + comp_fes_set = {face_edges_set(fd[0]) + for _, fd in comp} + vertices = set() + for fes in comp_fes_set: + for e in fes: + vertices.update(e) + for v in vertices: + rot = emb[v] + in_c = [] + for i in range(len(rot)): + u = rot[i] + w = rot[(i + 1) % len(rot)] + tri = frozenset({frozenset({v, u}), + frozenset({u, w}), + frozenset({v, w})}) + in_c.append(tri in comp_fes_set) + n_true = sum(in_c) + if n_true == 0 or n_true == len(in_c): + continue + n_runs = sum( + 1 for i in range(len(in_c)) + if in_c[i] and not in_c[(i - 1) % len(in_c)]) + if n_runs > 1: + # Found one + print(f"=== R1 violation candidate ===") + print(f"n={n}, source={v_source}, " + f"depth={target_d}, pinch vertex={v}") + print(f"edges of G: " + f"{sorted([tuple(sorted(e)) for e in G.edges(labels=False)])}") + print(f"BFS distances: {dict(dist)}") + print(f"rotation around v={v}: {rot}") + lvls = [dist[u] for u in rot] + print(f" levels: {lvls}") + face_depths_around_v = [] + for i in range(len(rot)): + u = rot[i] + w = rot[(i + 1) % len(rot)] + d_face = min(dist[v], dist[u], dist[w]) + face_depths_around_v.append(d_face) + print(f" face depths: {face_depths_around_v}") + print(f" in_component: {in_c}") + print(f" n_runs of True: {n_runs}") + # Also list the depth-d faces at v + d_target_at_v = [ + (rot[i], rot[(i + 1) % len(rot)]) + for i in range(len(rot)) + if face_depths_around_v[i] == target_d + ] + print(f" depth-{target_d} faces at v: " + f"{d_target_at_v}") + return + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/classify_r1_pinches.py b/papers/coloring_nested_tire_graphs/experiments/classify_r1_pinches.py new file mode 100644 index 0000000..7990b97 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/classify_r1_pinches.py @@ -0,0 +1,146 @@ +"""For each claimed (R1) pinch vertex, classify it as: + (a) a cut-vertex of O = G[L_{d+1} cap V_{C'}] (level d+1 pinch) + (b) a cut-vertex of G[L_d cap V_{C'}] (level d pinch) + (c) other +""" +import sys +from collections import defaultdict + +from sage.all import Graph +from sage.graphs.graph_generators import graphs + + +def face_edges_set(face): + return frozenset(frozenset(e) for e in face) + + +def is_cut_vertex(G, v): + """Return True if removing v disconnects the connected component of v.""" + if v not in G: + return False + H = G.copy() + H.delete_vertex(v) + if H.order() == 0: + return False + return not H.is_connected() + + +def main(): + counts = defaultdict(int) + n_components = 0 + n_pinches = 0 + sample_each = {} + for n in range(7, 12): + for G in graphs.triangulations(n): + G.is_planar(set_embedding=True) + emb = G.get_embedding() + faces = G.faces(embedding=emb) + for v_source in G.vertices(): + dist = G.shortest_path_lengths(v_source) + face_depths = [] + for face in faces: + verts = set() + for u, v in face: + verts.add(u); verts.add(v) + d = min(dist[u] for u in verts) + face_depths.append((face, d, verts)) + max_d = max(fd[1] for fd in face_depths) + for target_d in range(max_d + 1): + cands = [(i, fd) for i, fd in enumerate(face_depths) + if fd[1] == target_d] + if not cands: + continue + edge_to_idxs = defaultdict(list) + for k, (i, fd) in enumerate(cands): + for e in [frozenset(ed) for ed in fd[0]]: + edge_to_idxs[e].append(k) + n_c = len(cands) + adj = [[] for _ in range(n_c)] + for idxs in edge_to_idxs.values(): + if len(idxs) >= 2: + for a in idxs: + for b in idxs: + if a != b: + adj[a].append(b) + visited = [False] * n_c + comps = [] + for s in range(n_c): + if visited[s]: + continue + stack = [s]; comp = [] + while stack: + x = stack.pop() + if visited[x]: continue + visited[x] = True; comp.append(cands[x]) + for y in adj[x]: + if not visited[y]: stack.append(y) + comps.append(comp) + + for comp in comps: + n_components += 1 + comp_fes_set = {face_edges_set(fd[0]) + for _, fd in comp} + vertices_in_comp = set() + for fes in comp_fes_set: + for e in fes: + vertices_in_comp.update(e) + Vd = {v for v in vertices_in_comp if dist[v] == target_d} + Vdp1 = {v for v in vertices_in_comp if dist[v] == target_d + 1} + O = G.subgraph(vertices=list(Vdp1)) + Bout_graph = G.subgraph(vertices=list(Vd)) + for v in vertices_in_comp: + rot = emb[v] + in_c = [] + for i in range(len(rot)): + u = rot[i] + w = rot[(i + 1) % len(rot)] + tri = frozenset({frozenset({v, u}), + frozenset({u, w}), + frozenset({v, w})}) + in_c.append(tri in comp_fes_set) + n_true = sum(in_c) + if n_true == 0 or n_true == len(in_c): + continue + n_runs = sum( + 1 for i in range(len(in_c)) + if in_c[i] and not in_c[(i - 1) % len(in_c)]) + if n_runs <= 1: + continue + # Pinch at v. Classify. + n_pinches += 1 + level_v = dist[v] + if level_v == target_d: + lvl_label = 'd' + # Check if v is a cut vertex of B_out_graph + cut_bout = is_cut_vertex(Bout_graph, v) + key = ('d', cut_bout) + elif level_v == target_d + 1: + lvl_label = 'd+1' + cut_o = is_cut_vertex(O, v) + key = ('d+1', cut_o) + else: + key = ('other', None) + counts[key] += 1 + if key not in sample_each: + sample_each[key] = { + 'n': n, 'source': int(v_source), + 'depth': target_d, 'pinch_v': v, + 'level_v': level_v, + 'edges': sorted(tuple(sorted(e)) + for e in G.edges(labels=False)), + } + + print(f"Total components: {n_components}") + print(f"Total pinches: {n_pinches}") + print() + print("Classification by (level of pinch, is cut-vertex of corresponding side):") + for k, c in sorted(counts.items()): + print(f" {k}: {c}") + print() + print("Samples:") + for k, sample in sample_each.items(): + print(f" key={k}: {sample}") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_r1_candidate.py b/papers/coloring_nested_tire_graphs/experiments/draw_r1_candidate.py new file mode 100644 index 0000000..c1d8039 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_r1_candidate.py @@ -0,0 +1,172 @@ +"""Draw the claimed (R1) violation: n=9, source=5, depth=1, vertex 9. + +The depth-1 component has 9 triangles, and vertex 9 has 4 of them +incident (in two arcs of the rotation around 9, separated by 2 +depth-2 faces -- the two triangles of the bowtie at L_2). + +The question: is vertex 9 a real "pinch" of the tire, or is it just +a cut-vertex of the inner outerplanar graph O (= the bowtie on +{2, 3, 7, 8, 9}), naturally handled by the relaxed Definition 1.5? +""" +import math +import os +import sys +from collections import defaultdict + +from sage.all import Graph +import matplotlib.pyplot as plt +import matplotlib.patches as patches + +HERE = os.path.dirname(os.path.abspath(__file__)) + +EDGES = [(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), + (2, 3), (2, 9), (3, 4), (3, 9), (4, 5), (4, 6), (4, 7), (4, 9), + (5, 6), (6, 7), (7, 8), (7, 9), (8, 9)] +SOURCE = 5 + + +def main(): + G = Graph(EDGES) + G.is_planar(set_embedding=True) + emb = G.get_embedding() + dist = G.shortest_path_lengths(SOURCE) + + # Pick an external face that doesn't contain source 5 so source + # ends up in the interior. + # Faces of G that don't contain 5: triangle {1,2,3} for example. + pos = G.layout(layout='planar', external_face=(1, 2)) + # Normalise + xs = [p[0] for p in pos.values()]; ys = [p[1] for p in pos.values()] + cx, cy = (max(xs)+min(xs))/2, (max(ys)+min(ys))/2 + scale = max(max(xs)-min(xs), max(ys)-min(ys)) + pos = {v: ((p[0]-cx)/scale, (p[1]-cy)/scale) for v, p in pos.items()} + + faces = G.faces(embedding=emb) + face_depths = [] + for face in faces: + verts = set() + for u, v in face: + verts.add(u); verts.add(v) + d = min(dist[u] for u in verts) + face_depths.append((face, d, verts)) + depth1_faces = [fd for fd in face_depths if fd[1] == 1] + depth2_faces = [fd for fd in face_depths if fd[1] == 2] + + fig, ax = plt.subplots(figsize=(10, 9)) + + # Fill depth-1 faces (R_{C'}) + for face, _, _ in depth1_faces: + verts = [e[0] for e in face] + xy = [pos[v] for v in verts] + poly = plt.Polygon(xy, facecolor='#fff3cd', edgecolor='#e8d8a8', + linewidth=0.5, alpha=0.85, zorder=1) + ax.add_patch(poly) + + # Fill depth-2 faces (the bowtie's two triangles) in a different colour + for face, _, _ in depth2_faces: + verts = [e[0] for e in face] + xy = [pos[v] for v in verts] + poly = plt.Polygon(xy, facecolor='#d62728', edgecolor='#a01818', + linewidth=0.5, alpha=0.40, zorder=1) + ax.add_patch(poly) + + # All edges of G in grey + for u, v in G.edges(labels=False): + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color='lightgray', linewidth=1.0, + zorder=2) + + # Highlight the bowtie edges (= edges of O = G[L_2]) + L2 = [v for v in G.vertices() if dist[v] == 2] + print(f"L_2 = {sorted(L2)}") + bowtie_edges = [(u, v) for u, v in G.edges(labels=False) + if u in L2 and v in L2] + print(f"Edges of O = G[L_2]: {bowtie_edges}") + for u, v in bowtie_edges: + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color='#a01818', linewidth=3.0, + zorder=3) + + # Highlight the source-side cycle = G[L_1] + L1 = [v for v in G.vertices() if dist[v] == 1] + print(f"L_1 = {sorted(L1)}") + L1_edges = [(u, v) for u, v in G.edges(labels=False) + if u in L1 and v in L1] + print(f"Edges of B_out candidate = G[L_1]: {L1_edges}") + for u, v in L1_edges: + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color='#1f77b4', linewidth=3.0, + zorder=3) + + # Vertices + level_colors = {0: 'black', 1: '#1f77b4', 2: '#d62728'} + for v in G.vertices(): + x, y = pos[v] + c = level_colors.get(dist[v], 'gray') + ax.plot(x, y, 'o', color=c, markersize=22, zorder=4) + ax.annotate(f"{v}", (x, y), color='white', ha='center', va='center', + fontsize=11, fontweight='bold', zorder=5) + + # Special annotation: vertex 9 is the cut-vertex of O + if 9 in pos: + x, y = pos[9] + ax.annotate('cut-vertex of $O$', + xy=(x, y), xytext=(x + 0.10, y + 0.18), + fontsize=10, color='#a01818', fontweight='bold', + arrowprops=dict(arrowstyle='->', color='#a01818', lw=1.5), + zorder=6, + bbox=dict(boxstyle='round,pad=0.25', + facecolor='white', edgecolor='#a01818', + alpha=0.95)) + + if SOURCE in pos: + sx, sy = pos[SOURCE] + ax.annotate(f'source $v_0={SOURCE}$', + xy=(sx, sy), xytext=(sx + 0.08, sy + 0.10), + fontsize=10, color='black', fontweight='bold', + arrowprops=dict(arrowstyle='->', color='black', lw=1.2), + zorder=6, + bbox=dict(boxstyle='round,pad=0.2', + facecolor='white', edgecolor='gray', alpha=0.95)) + + legend_items = [ + plt.Line2D([], [], marker='o', color='w', markerfacecolor='black', + markersize=12, label=f'source ($\\ell=0$)'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#1f77b4', + markersize=12, label='$\\ell=1$'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#d62728', + markersize=12, label='$\\ell=2$'), + patches.Patch(facecolor='#fff3cd', edgecolor='#e8d8a8', + label='$R_{C^\\prime}$ (depth-1 component, ' + '9 triangles)'), + patches.Patch(facecolor='#d62728', alpha=0.4, + label='depth-2 faces (interior faces of $O$)'), + plt.Line2D([], [], color='#1f77b4', linewidth=3.0, + label='$B_{\\mathrm{out}} = G[L_1]$'), + plt.Line2D([], [], color='#a01818', linewidth=3.0, + label='$O = G[L_2]$ = bowtie with cut-vertex 9'), + ] + ax.legend(handles=legend_items, loc='upper right', fontsize=9, + framealpha=0.95) + + ax.set_title( + '"(R1) candidate": vertex 9 is a cut-vertex of $O$ (the bowtie),\n' + 'not a true pinch of the tire under the relaxed Definition 1.5\n' + '(n=9, source $v_0=5$, depth 1)', + fontsize=11) + ax.set_aspect('equal') + ax.axis('off') + pad = 0.08 + ax.set_xlim(min(p[0] for p in pos.values()) - pad, + max(p[0] for p in pos.values()) + pad) + ax.set_ylim(min(p[1] for p in pos.values()) - pad, + max(p[1] for p in pos.values()) + pad) + + out = os.path.join(HERE, 'r1_candidate.png') + plt.savefig(out, dpi=160, bbox_inches='tight') + plt.close() + print(f"wrote {out}") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/r1_candidate.png b/papers/coloring_nested_tire_graphs/experiments/r1_candidate.png new file mode 100644 index 0000000000000000000000000000000000000000..5d1cffa936a24d2f44c1152953db9d49536e45b4 GIT binary patch literal 111391 zcmce;bx@S=+czvorzo8Q3RpDKDPSU?q_lK*Ei5Uav;op7%~H~_0tzVI4Z;Eo?2=0_ z@tphp{l51+Gw=J)J@3rRjH801FBWZtLtM!Y?GqFLwW? zjfaPeyOe-{U%A4Viu;B4)HC$+6}&4~o+>M9`J|!e zNSj|AV2O8-TMQGe0#jj6?%cM#^N~JBc{nG&O#A7pocPZVH(%X)mi;P#lTP#L?dKK? zM^I$r%n4l11d5dR-jb}&(BDR$Lt+}N`qR$SB$3HTtUrRHL-1YXf4}&Men$8a{rk5_ zbh{}7&40gsQ4lr#`vX2J!ATY6|9%ODw()cQ_v-@_h1b6?(Cjs>i2C;xZ;3*U{(S+h z2qXI67vy|x`~T%FG#L+26Y5J#OBNfWMKpD6%;HY>Nl8hoAO{;R1J9I;-qaLNTixJ& 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triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{2}} \newlabel{def:tire-graph}{{1.5}{2}} +\citation{bauerfeld-pds} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{3}} \newlabel{rem:tire-counts}{{1.6}{3}} \newlabel{lem:tire-component}{{1.7}{3}} \citation{bauerfeld-pds} -\citation{bauerfeld-pds} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} @@ -17,6 +17,6 @@ \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} \newlabel{rem:tire-component-degenerate}{{1.8}{5}} -\newlabel{rem:R1-when}{{1.9}{5}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.9}{5}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } \gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 95b8b1e..4e088e5 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 16:26 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 16:37 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -205,13 +205,13 @@ LaTeX Warning: `h' float specifier changed to `ht'. 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graph, and its outer-face boundary -closed walk serves as $B_{\mathrm{in}}$. In particular, two disjoint -$L_{d+1}$ triangles inside $R_{C'}$ joined by a bridge edge in $O$ -give a $B_{\mathrm{in}}$ that traverses the bridge twice and visits -the bridge endpoints twice each --- not a simple cycle, but a -well-defined closed walk on $V(O)$. +\emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may +have the faces of $F_{C'}$ incident to it split into two or more +arcs in $v$'s rotation in $\Pi_G$, separated by faces of higher +depth. This corresponds exactly to $v$ being a cut-vertex of +$O = G[V_{C'} \cap L_{d+1}]$, and the inner boundary closed walk +$B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each +arc. No additional hypothesis is needed. -In the special case $d = 0$ with single-vertex source $S = \{v_0\}$ -(R1) holds automatically: $R_{C'}$ is the star of $v_0$, a topological -closed disk with one boundary cycle (the link of $v_0$), giving a -tire graph with degenerate outer boundary $\{v_0\}$. +\emph{Multi-hole topology of $R_{C'}$.} Even when $R_{C'}$ encloses +several disjoint depth-$>d$ sub-regions, the inner outerplanar graph +$O$ captures the multi-hole structure as a disconnected or +non-$2$-connected outerplanar graph (with bridges or multiple +components), and its outer-face boundary closed walk serves as +$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices +multiple times. + +In the special case $d = 0$ with single-vertex source $S = \{v_0\}$, +$R_{C'}$ is the star of $v_0$, a topological closed disk with one +boundary cycle (the link of $v_0$); the corresponding tire graph has +degenerate outer boundary $\{v_0\}$. \end{remark} \begin{thebibliography}{9}