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math-research/papers/level_switching/experiments/make_definition_figures.py
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didericis 7183dc1b67 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>
2026-05-20 23:08:22 -04:00

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"""Generate the three definition figures for the Level Switching paper.
Uses a stacked 7-vertex triangulation T:
outer triangle {0,1,2}, inner vertex 3 connected to all three,
then vertices 4,5,6 inserted into faces (1,2,3),(0,2,3),(0,1,3).
"""
import os
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)
# Vertex positions (hand-placed for a clean planar drawing).
POS = {
0: (-1.5, -0.9),
1: (1.5, -0.9),
2: (0.0, 1.6),
3: (0.0, 0.0),
4: (0.55, 0.2), # in face (1,2,3)
5: (-0.55, 0.2), # in face (0,2,3)
6: (0.0, -0.55), # in face (0,1,3)
}
EDGES = [
(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), # K_4
(4, 1), (4, 2), (4, 3), # stack in (1,2,3)
(5, 0), (5, 2), (5, 3), # stack in (0,2,3)
(6, 0), (6, 1), (6, 3), # stack in (0,1,3)
]
def make_graph():
G = nx.Graph()
G.add_nodes_from(POS.keys())
G.add_edges_from(EDGES)
return G
def draw_base(ax, G, node_colors, node_size=520, font_color='white',
edge_color='#555', edge_width=1.4):
nx.draw_networkx_edges(G, POS, ax=ax, edge_color=edge_color, width=edge_width)
nx.draw_networkx_nodes(G, POS, ax=ax, node_color=node_colors,
node_size=node_size, edgecolors='black', linewidths=1.0)
nx.draw_networkx_labels(G, POS, ax=ax, font_color=font_color,
font_size=11, font_weight='bold')
ax.set_aspect('equal')
ax.axis('off')
# ---------------------------------------------------------------------------
# Figure 1: Level source (face source vs. degree-3 vertex source)
# ---------------------------------------------------------------------------
def fig_level_source():
G = make_graph()
fig, axes = plt.subplots(1, 2, figsize=(10, 5))
# Panel A: face source S = {0,1,2}
ax = axes[0]
face_S = {0, 1, 2}
colors = ['#ef4444' if v in face_S else '#cbd5e1' for v in G.nodes()]
# Highlight the source face
tri = Polygon([POS[v] for v in [0, 1, 2]], closed=True,
facecolor='#fecaca', edgecolor='#ef4444', linewidth=2.0,
alpha=0.45, zorder=0)
ax.add_patch(tri)
draw_base(ax, G, colors)
ax.set_title(r'Face source $S = \{0,1,2\}$', fontsize=12)
# Panel B: degree-3 vertex source S = {4}
ax = axes[1]
vert_S = {4}
colors = ['#ef4444' if v in vert_S else '#cbd5e1' for v in G.nodes()]
draw_base(ax, G, colors)
ax.set_title(r'Degree-3 vertex source $S = \{4\}$', fontsize=12)
fig.tight_layout()
out = os.path.join(OUT_DIR, 'fig_level_source.png')
fig.savefig(out, dpi=200, bbox_inches='tight')
plt.close(fig)
print(f'wrote {out}')
# ---------------------------------------------------------------------------
# Figure 2: Levels (BFS distance from a source)
# ---------------------------------------------------------------------------
def fig_levels():
G = make_graph()
source = 4 # degree-3 vertex source
levels = nx.single_source_shortest_path_length(G, source)
# Color by level
palette = {0: '#ef4444', 1: '#f59e0b', 2: '#3b82f6'}
colors = [palette[levels[v]] for v in G.nodes()]
# Labels = level numbers
labels = {v: rf'$\ell={levels[v]}$' for v in G.nodes()}
fig, ax = plt.subplots(figsize=(6.5, 5.5))
nx.draw_networkx_edges(G, POS, ax=ax, edge_color='#555', width=1.4)
nx.draw_networkx_nodes(G, POS, ax=ax, node_color=colors,
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()
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