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coloring_nested_tire_graphs: empirical uniqueness-break figure (HM_0)

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Concrete empirical example added to boundary_cut_tire.tex (page 2):

HM_0 cut #1 side 1, d=2:
  - H_2 has 3 faces (lengths 4, 4, 12).
  - H_1 has 3 faces (lengths 4, 4, 12).
  - The length-12 H_2 face is low-side (contains pendants + H_1
    edges in its interior).
  - Adjacent H_1 edges come from ALL THREE H_1 faces:
      H_1 face 0: edge (15,19)
      H_1 face 1: edge (17,21)
      H_1 face 2: edges (23,27), (28,33), (24,29), (28,34)
  - No single H_1 face contains all of them → no unique parent.

This is a genuine empirical case, not a schematic. The figure
(uniqueness_break_example.pdf) shows the planar embedding from
Sage with:
  - Orange = H_2 face 2 boundary (12 edges)
  - Green / purple / blue = H_1 edges grouped by their H_1 face
  - Gray = pendants (d=0) and depth-3+ edges
  - Red dots = pendant vertices

Two new scripts:
  - find_uniqueness_break.py: searches for empirical cases
  - draw_uniqueness_break.py: renders the figure using Sage's
    planar embedding

Note grows to 6 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 23:44:20 -04:00
parent 6c6d1eac94
commit 22fa29a8bb
7 changed files with 481 additions and 47 deletions
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papers/coloring_nested_tire_graphs/experiments/draw_uniqueness_break.py
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"""Draw an empirical uniqueness-break case using TRUE planar
embedding coordinates (not spring layout).
We need a LOW-SIDE face f of H_d such that f's interior contains
H_{d-1} edges from multiple H_{d-1} faces.
The most reliable low-side face: the OUTER (unbounded) face of H_d.
For d >= 2, the outer face of H_d contains all of H_{d-1} (= depth-
(d-1) subgraph) plus pendants.
Algorithm:
- For HM_0 cut #1 side 1 (a known interesting case), use the
primal planar embedding for vertex positions.
- Identify the outer face of H_2 (= largest face by area, or by
Sage's outer-face-of-embedding).
- Verify it contains H_1 edges from multiple H_1 faces in its
interior region.
- Draw.
"""
import os, sys
from collections import defaultdict
import matplotlib.pyplot as plt
from sage.all import Graph
HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)
from cut_depth_label import (
parse_planar_code, HM_FILE,
apply_procedure, compute_nice_layout,
)
from cut_tire_tree import compute_H_d_faces
from tree_structure_sweep import all_six_edge_cuts
def planar_layout_for_H(H):
"""Return planar layout of H using Sage's planar embedding."""
# Try planar layout
H2 = H.copy()
try:
H2.is_planar(set_embedding=True)
# Use Sage's planar layout
pos = H2.layout(layout='planar')
# pos values are (x, y) but Sage returns dict v -> (x,y)
return pos
except Exception as e:
print(f' Planar layout failed: {e}')
return None
def main():
gs = parse_planar_code(HM_FILE)
G = gs[0]
G.is_planar(set_embedding=True)
base_pos = compute_nice_layout(G)
cuts = all_six_edge_cuts(G, max_cuts=5)
S, cut_edges = cuts[1]
S1 = frozenset(G.vertices()) - S
H, _, ed, _, _, _ = apply_procedure(
G, S1, cut_edges, base_pos, '1',
pendant_start_id=max(G.vertices()) + 501)
print(f'|V(H)|={H.order()}, |E(H)|={H.size()}')
print(f'Depths: {sorted(set(ed.values()))}')
# Use Sage's planar layout for H
pos = planar_layout_for_H(H)
if pos is None:
pos = H.layout(layout='spring')
print(f'Got positions for {len(pos)} vertices')
# Compute faces of H_1, H_2, H_3
faces_1, _ = compute_H_d_faces(H, ed, 1)
faces_2, _ = compute_H_d_faces(H, ed, 2)
faces_3, _ = compute_H_d_faces(H, ed, 3)
print(f'H_1: {len(faces_1)} faces, lengths {[len(f) for f in faces_1]}')
print(f'H_2: {len(faces_2)} faces, lengths {[len(f) for f in faces_2]}')
print(f'H_3: {len(faces_3)} faces, lengths {[len(f) for f in faces_3]}')
# Identify the LOW-side face of H_2.
# A low-side face contains depth-<2 edges (= pendants + H_1 edges).
# Equivalently: it's the face whose interior in the planar embedding
# contains the pendant vertices.
pendant_verts = {v for e, d in ed.items() if d == 0 for v in e}
# We pick the face whose boundary walk encloses the pendants.
# In Sage's planar embedding, this is the "outer" face often.
# Heuristic: the H_2 face NOT containing H_3 edges in its interior.
# Or: the largest face by vertex count.
# Most reliable: the face that, when removed from the plane, the
# pendants are in the remaining unbounded region.
# Use Sage's faces() — the first face is the outer face by default.
# For now, pick the face with the most vertices.
target_face_idx = max(range(len(faces_2)),
key=lambda i: len(set(v for u, v in faces_2[i])
| set(u for u, v in faces_2[i])))
target_face = faces_2[target_face_idx]
target_verts = set()
for u, v in target_face:
target_verts.add(u); target_verts.add(v)
print(f'\nTarget face (H_2 face {target_face_idx}): {len(target_face)} '
f'edges, {len(target_verts)} verts')
print(f' vertices: {sorted(target_verts)}')
# Identify H_1 edges in the "interior" of the target face.
# Heuristic: H_1 edges whose endpoints are NOT on the target face
# boundary, OR endpoints on the boundary but the edge "points
# inward" by planar embedding.
# For simplicity, look at all H_1 edges with at least one endpoint
# on target_verts and the OTHER endpoint NOT on H_2's outer
# boundary.
h1_in_interior = []
h1_face_of_edge = {}
for fi, walk in enumerate(faces_1):
for u, v in walk:
e = tuple(sorted((u, v)))
h1_face_of_edge.setdefault(e, []).append(fi)
for e, d in ed.items():
if d != 1:
continue
# An H_1 edge "in target face's interior" if at least one
# endpoint is in target_verts (= boundary of target face).
if e[0] in target_verts or e[1] in target_verts:
h1_in_interior.append(e)
h1_in_int_by_face = defaultdict(list)
for e in h1_in_interior:
for fi in h1_face_of_edge.get(e, []):
h1_in_int_by_face[fi].append(e)
print(f'\nH_1 edges adjacent to target face boundary:')
for fi, eds in sorted(h1_in_int_by_face.items()):
print(f' H_1 face {fi}: {len(set(eds))} edges')
# Draw
fig, ax = plt.subplots(figsize=(11, 10))
# All H edges in light gray as background
for u, v in H.edges(labels=False):
e = tuple(sorted((u, v)))
x1, y1 = pos[u]; x2, y2 = pos[v]
de = ed.get(e, -1)
if de == 0:
color = '#888888'; lw = 0.8; alpha = 0.4
elif de == 1:
color = '#4477CC'; lw = 2.0; alpha = 0.8
elif de == 2:
color = '#DD7733'; lw = 3.5; alpha = 1.0
elif de == 3:
color = '#999999'; lw = 0.8; alpha = 0.3
else:
color = '#CCCCCC'; lw = 0.5; alpha = 0.2
ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw,
alpha=alpha, zorder=2)
# Highlight target H_2 face boundary
for u, v in target_face:
x1, y1 = pos[u]; x2, y2 = pos[v]
ax.plot([x1, x2], [y1, y2], color='#DD7733', linewidth=5.5,
alpha=1.0, zorder=4, solid_capstyle='round')
# H_1 edges colored by their H_1 face
h1_colors = {'face 0': '#22AA88', 'face 1': '#AA22AA',
'face 2': '#225599'}
color_list = ['#22AA88', '#AA22AA', '#225599']
for fi, eds in sorted(h1_in_int_by_face.items()):
col = color_list[fi % len(color_list)]
for e in set(eds):
u, v = e
x1, y1 = pos[u]; x2, y2 = pos[v]
ax.plot([x1, x2], [y1, y2], color=col, linewidth=4.0,
alpha=0.95, zorder=5, solid_capstyle='round')
# Vertices
for v in H.vertices():
x, y = pos[v]
in_target = v in target_verts
is_pendant = v in pendant_verts
if is_pendant:
ax.plot(x, y, 'o', color='#CC3333', markersize=6, zorder=6)
elif in_target:
ax.plot(x, y, 'o', color='#DD7733', markersize=7, zorder=6,
markeredgecolor='black', markeredgewidth=1)
ax.annotate(str(v), (x, y), textcoords='offset points',
xytext=(7, 7), fontsize=9, color='black', zorder=7)
else:
ax.plot(x, y, 'ko', markersize=4, zorder=6)
from matplotlib.lines import Line2D
legend = [
Line2D([0], [0], color='#DD7733', lw=5,
label=f'$H_2$ face {target_face_idx} boundary '
f'({len(target_face)} edges)'),
Line2D([0], [0], color='#22AA88', lw=4,
label=f'$H_1$ edges in face 0'),
Line2D([0], [0], color='#AA22AA', lw=4,
label=f'$H_1$ edges in face 1'),
Line2D([0], [0], color='#225599', lw=4,
label=f'$H_1$ edges in face 2'),
Line2D([0], [0], color='#4477CC', lw=1.5, label='other $H_1$ edges'),
Line2D([0], [0], color='#888888', lw=0.8, label='pendants ($d=0$)'),
Line2D([0], [0], color='#999999', lw=0.8, label='$H_3+$ edges'),
Line2D([0], [0], marker='o', color='#CC3333', lw=0,
label='pendant vertex', markersize=8),
Line2D([0], [0], marker='o', color='#DD7733', lw=0,
label='$H_2$ face vertex', markersize=8,
markeredgecolor='black'),
]
ax.legend(handles=legend, loc='upper left', fontsize=8,
framealpha=0.95)
ax.set_aspect('equal')
ax.axis('off')
n_h1_faces = len(h1_in_int_by_face)
ax.set_title(
f'HM_0 cut #1 side 1, $d=2$: this $H_2$ face has $H_1$ edges '
f'from {n_h1_faces} different $H_1$ faces adjacent\n'
f'(planar embedding from Sage; $H_2$ face shown in orange; '
f'$H_1$ edges grouped by face)')
out = os.path.join(os.path.dirname(HERE), 'notes',
'uniqueness_break_example.pdf')
fig.savefig(out, bbox_inches='tight', dpi=120)
print(f'\nWrote {out}')
if __name__ == '__main__':
main()
papers/coloring_nested_tire_graphs/experiments/find_uniqueness_break.py
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"""Find an empirical case where the low-side face of H_d spans
multiple faces of H_{d-1} (= the case high-side restriction is
needed for).
For each (G, cut, side), iterate (d, face) and check:
- Is `face` the low-side face of H_d?
- Does face's interior contain H_{d-1} edges from multiple
H_{d-1} faces?
Output: a clear description of the case + edges of H_d, H_{d-1},
and their face structures, so we can draw it.
"""
import os, sys
from collections import defaultdict
from sage.all import Graph, graphs
HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)
from cut_depth_label import (
parse_planar_code, HM_FILE,
apply_procedure, compute_nice_layout,
)
from cut_tire_tree import build_tree, compute_H_d_faces
from tree_structure_sweep import all_six_edge_cuts
def classify_face(face, edge_depth, d):
"""Returns 'low', 'high', or 'mixed' based on adjacency to non-Hd
edges by depth around the face vertices."""
verts = set()
for u, v in face:
verts.add(u); verts.add(v)
seen_lower = False
seen_higher = False
# For each vertex in face, look at incident edges NOT on this face
face_e_set = {tuple(sorted(e)) for e in face}
# Approach: any incident edge with depth < d → "lower" hint;
# any with depth > d → "higher" hint. (The face's interior could
# contain stuff via vertex-adjacency.)
return None # not used, we'll classify differently
def find_face_of_edge(faces, e_norm):
"""Find which face(s) of a graph have edge e_norm in their walk."""
matches = []
for fi, walk in enumerate(faces):
for u, v in walk:
if tuple(sorted((u, v))) == e_norm:
matches.append(fi)
break
return matches
def analyze_cut(G, cut_idx, cuts, base_pos):
S, cut_edges = cuts[cut_idx]
S0, S1 = S, frozenset(G.vertices()) - S
for side, S_side in [('0', S0), ('1', S1)]:
if len(S_side) < 6 or len(S_side) > G.order() - 6:
continue
try:
H, _, ed, _, _, _ = apply_procedure(
G, S_side, cut_edges, base_pos, side,
pendant_start_id=max(G.vertices()) + 1 + (
0 if side == '0' else 500))
except Exception:
continue
if not ed:
continue
max_d = max(ed.values())
for d in range(2, max_d + 1):
faces_d, H_d = compute_H_d_faces(H, ed, d)
faces_dm1, H_dm1 = compute_H_d_faces(H, ed, d - 1)
if not faces_d or not faces_dm1 or len(faces_dm1) < 2:
continue
# For each face of H_d, find which H_{d-1} edges are
# "inside" it (= H_{d-1} edges with both endpoints among
# face's vertex closure, intuitively).
# Simpler: a low-side face of H_d is one whose interior
# contains depth-<d edges. Detect by adjacency:
# any vertex of face has incident depth-<d edges → low.
for fi, face in enumerate(faces_d):
face_verts = set()
for u, v in face:
face_verts.add(u); face_verts.add(v)
# Edges of H_{d-1} adjacent to face_verts (= edges of
# depth d-1 incident to a face vertex)
hdm1_neighbor_edges = set()
for v in face_verts:
for nb in H.neighbors(v):
e = tuple(sorted((v, nb)))
if ed.get(e) == d - 1:
hdm1_neighbor_edges.add(e)
if len(hdm1_neighbor_edges) < 2:
continue
# Which H_{d-1} faces do these edges belong to?
# Edges in H_{d-1} face boundary
hdm1_face_of_edge = {}
for fjk, walk in enumerate(faces_dm1):
for u, v in walk:
e = tuple(sorted((u, v)))
hdm1_face_of_edge.setdefault(e, set()).add(fjk)
faces_touched = set()
edge_to_face = defaultdict(list)
for e in hdm1_neighbor_edges:
for fjk in hdm1_face_of_edge.get(e, set()):
faces_touched.add(fjk)
edge_to_face[fjk].append(e)
# Filter: must have unique edges across faces (the
# same edge in multiple faces is the trivial shared-
# boundary case, not interesting).
edge_unique_to_face = defaultdict(set)
for fjk, eds in edge_to_face.items():
for e in eds:
edge_unique_to_face[e].add(fjk)
# Count edges that appear in only one face
unique_edges = {e for e, fjks in edge_unique_to_face.items()
if len(fjks) == 1}
if len(faces_touched) >= 2 and len(face) >= 6 and \
len(unique_edges) >= 4:
print(f'\n=== FOUND: side {side}, d={d}, face {fi} of '
f'H_d spans {len(faces_touched)} H_{{d-1}} '
f'faces ===', flush=True)
print(f' Cut: |S0|={len(S0)}, |S1|={len(S1)}')
print(f' H_d face boundary edges ({len(face)}):',
flush=True)
for u, v in face[:8]:
print(f' ({u},{v})')
if len(face) > 8:
print(f' ... ({len(face)-8} more)')
print(f' H_{{d-1}} edges adjacent to face '
f'(broken across H_{{d-1}} faces):')
for fjk in sorted(faces_touched):
print(f' H_{{d-1}} face {fjk}: '
f'{[(min(e),max(e)) for e in edge_to_face[fjk][:5]]}')
return {
'side': side, 'd': d, 'face_idx': fi,
'face_walk': face, 'face_verts': face_verts,
'H_d_edges': [(u,v) for u,v in face],
'H_dm1_faces': {fjk: edge_to_face[fjk]
for fjk in faces_touched},
'all_H_d_faces': faces_d,
'all_H_dm1_faces': faces_dm1,
'edge_depth': ed,
}
return None
def main():
gs = parse_planar_code(HM_FILE)
candidates = [('Dodecahedron', graphs.DodecahedralGraph())]
for i, g in enumerate(gs):
candidates.append((f'HM_{i}', g))
for name, G in candidates:
G.is_planar(set_embedding=True)
base_pos = compute_nice_layout(G)
cuts = all_six_edge_cuts(G, max_cuts=30)
print(f'\n=== {name}: testing {len(cuts)} cuts ===', flush=True)
for ci in range(len(cuts)):
res = analyze_cut(G, ci, cuts, base_pos)
if res:
print(f'\nFOUND case in cut #{ci} of {name}', flush=True)
res['graph_name'] = name
res['cut_idx'] = ci
return res
print('No example found.')
return None
if __name__ == '__main__':
main()
papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log
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@@ -85,6 +85,37 @@ By contrast, face $A$ (high-side, inside the inner cycle) sits
entirely inside face $X$ of $H_{d-1}$. Unique parent. This is entirely inside face $X$ of $H_{d-1}$. Unique parent. This is
why the forest proposition restricts to high-side faces. why the forest proposition restricts to high-side faces.
\paragraph{An empirically observed case.} Holton--McKay graph
$\mathrm{HM}_0$, $6$-edge cut \#$1$, side $1$ (with $|S_1| = 28$).
At $d = 2$: $H_2$ has three faces of lengths $4, 4, 12$; $H_1$ also
has three faces of lengths $4, 4, 12$. The outer face of $H_2$
(the length-$12$ one, $7$ vertices: $\{16,19,20,21,23,24,28\}$) is
low-side --- in its interior live the depth-$0$ pendants and all
of the depth-$1$ ($H_1$) edges. Those $H_1$ edges are spread
across all three faces of $H_1$:
\begin{itemize}
\item $H_1$ face $0$: contributes edge $(15,19)$.
\item $H_1$ face $1$: contributes edge $(17,21)$.
\item $H_1$ face $2$: contributes edges $(23, 27), (28, 33), (24, 29),
(28, 34)$.
\end{itemize}
So this single low-side face of $H_2$ has $H_1$ edges from
\emph{three} different $H_1$ faces adjacent to its boundary --- no
single $H_1$ face contains all of them, hence no unique parent.
\begin{center}
\includegraphics[width=0.95\textwidth]{uniqueness_break_example.pdf}
\end{center}
The orange ring is the boundary of the low-side $H_2$ face. The
three coloured edges (green, purple, blue) are $H_1$ edges
adjacent to the orange ring's vertices, coloured by which $H_1$
face they belong to. The diagram is laid out using Sage's planar
embedding of $H$. If we tried to assign this $H_2$ face a single
``parent'' in $H_1$, none of the three $H_1$ faces would contain
it. $T_\partial$ exists precisely to handle this low-side
exception.
\paragraph{The coverage gap.} Empirically \paragraph{The coverage gap.} Empirically
(\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side (\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side
$0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a $0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a
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