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>

papers/level_switching/experiments/preprocessing_demo.py

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+"""Demonstrate the preprocessing strategy on the 9-vertex example.
+
+Start: F = (0,3,6) at depth 1, no balanced surface switch exists
+(F has no edge of "span 1" -- no edge with a single outer-cycle
+vertex between the endpoints, hence no ear neighbour).
+
+Step: perform the (unbalanced) surface switch on edge uv = 03, with
+F' = (0,2,3) and third vertex x = 2; in Case (ii) the flip removes 03
+and adds wx = 62.
+
+Result: A = (0,2,6) at depth 1 has edge 02 at span 1, so the ear
+(0,1,2) is now a balanced-switch target.
+"""
+import os
+import math
+import networkx as nx
+import matplotlib.pyplot as plt
+from matplotlib.patches import Polygon
+
+OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
+
+n = 9
+POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
+           math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
+OUTER_EDGES = [(i, (i + 1) % n) for i in range(n)]
+outer_set = {frozenset(e) for e in OUTER_EDGES}
+
+
+def face_edges(f):
+    return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
+            frozenset((f[0], f[2]))}
+
+
+def compute_depths(faces):
+    D = nx.Graph()
+    D.add_nodes_from(range(len(faces)))
+    for i, fi in enumerate(faces):
+        for j, fj in enumerate(faces):
+            if i < j and face_edges(fi) & face_edges(fj):
+                D.add_edge(i, j)
+    B = [i for i, f in enumerate(faces)
+         if len(face_edges(f) & outer_set) >= 1]
+    return {i: min(nx.shortest_path_length(D, i, b) for b in B)
+            for i in range(len(faces))}
+
+
+CHORDS_BEFORE = [(0, 2), (0, 3), (3, 5), (3, 6), (0, 6), (6, 8)]
+FACES_BEFORE = [
+    (0, 1, 2), (0, 2, 3), (3, 4, 5), (3, 5, 6),
+    (6, 7, 8), (6, 8, 0), (0, 3, 6),
+]
+
+# After non-balanced switch on edge 03: remove 03, add 26
+CHORDS_AFTER = [c for c in CHORDS_BEFORE if set(c) != {0, 3}] + [(2, 6)]
+FACES_AFTER = [
+    (0, 1, 2), (3, 4, 5), (3, 5, 6),
+    (6, 7, 8), (6, 8, 0),
+    (0, 2, 6), (2, 3, 6),
+]
+
+
+def draw(ax, faces, chords, depth, title, highlight_edges=None,
+         green_edges=None):
+    palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
+    edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
+    for i, f in enumerate(faces):
+        d = depth[i]
+        poly = Polygon([POS[v] for v in f], closed=True,
+                       facecolor=palette.get(d, '#ddd'),
+                       edgecolor=edge_pal.get(d, '#333'),
+                       linewidth=1.4, alpha=0.7, zorder=0)
+        ax.add_patch(poly)
+        cx = sum(POS[v][0] for v in f) / 3
+        cy = sum(POS[v][1] for v in f) / 3
+        ax.text(cx, cy, str(d), ha='center', va='center', fontsize=11,
+                color=edge_pal.get(d, '#333'), fontweight='bold')
+    for (a, b) in OUTER_EDGES + chords:
+        color = '#333'; lw = 1.2
+        if highlight_edges and ((a, b) in highlight_edges or
+                                 (b, a) in highlight_edges):
+            color = '#dc2626'; lw = 3.0
+        if green_edges and ((a, b) in green_edges or
+                            (b, a) in green_edges):
+            color = '#16a34a'; lw = 3.0
+        ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
+                color=color, linewidth=lw, zorder=1)
+    for i, (x, y) in POS.items():
+        ax.scatter([x], [y], s=270, c='#1f2937', edgecolors='black',
+                   linewidths=1.0, zorder=2)
+        ax.text(x, y, str(i), ha='center', va='center',
+                fontsize=9, color='white', fontweight='bold', zorder=3)
+    ax.set_aspect('equal'); ax.axis('off')
+    ax.set_xlim(-1.3, 1.3); ax.set_ylim(-1.3, 1.3)
+    ax.set_title(title, fontsize=11)
+
+
+depth_before = compute_depths(FACES_BEFORE)
+depth_after = compute_depths(FACES_AFTER)
+print('BEFORE:')
+for i, f in enumerate(FACES_BEFORE):
+    print(f'  {f} -> depth {depth_before[i]}')
+print('AFTER:')
+for i, f in enumerate(FACES_AFTER):
+    print(f'  {f} -> depth {depth_after[i]}')
+
+fig, axes = plt.subplots(1, 2, figsize=(14, 7))
+draw(axes[0], FACES_BEFORE, CHORDS_BEFORE, depth_before,
+     'Before: F=(0,3,6) depth 1; spans (2,2,2) so no ear neighbour',
+     highlight_edges=[(0, 3)])
+draw(axes[1], FACES_AFTER, CHORDS_AFTER, depth_after,
+     'After non-balanced switch 03->26: A=(0,2,6) depth 1; edge 02 has span 1',
+     green_edges=[(2, 6)], highlight_edges=[(0, 2)])
+
+fig.tight_layout()
+out = os.path.join(OUT_DIR, 'fig_preprocessing.png')
+fig.savefig(out, dpi=180, bbox_inches='tight')
+plt.close(fig)
+print(f'wrote {out}')