Adds a new note describing a cut-and-depth-label procedure for
the dual G' of a maximal planar G:
1. Find a 6-edge cut C in G'.
2. Remove cut edges → G'_0, G'_1.
3. In each G'_i:
a. V_i = degree-2 vertices (vertices incident to exactly 1
cut edge, hence degree 3-1=2 in induced subgraph).
b. For each v ∈ V_i, add a pendant edge to a new vertex.
Label pendants depth 0.
c. BFS-propagate: edges adjacent to a depth-d edge get
depth d+1, until all edges are labelled.
Worked example on Holton-McKay graph #0 (38-vertex non-Hamiltonian
cubic plane graph, dual of a 21-vertex triangulation):
- 128 distinct 6-edge cuts found by greedy search.
- Best matching cut: |S| = 10, cut = 6 edges with 12 distinct
endpoints (6 per side).
- G'_0: 10 + 6 = 16 vertices, max depth 2.
- G'_1: 28 + 6 = 34 vertices, max depth 7.
The procedure mirrors the 4CT cut-and-reglue reducibility scheme:
each G'_i has pendants restoring cubicity at the boundary; the
depth labels organize G'_i into concentric layers by distance to
the cut. This is the dual analogue of plane depth from a level
cycle (cf. the level-cycle generalization discussion).
Files:
experiments/cut_depth_label.py
notes/cut_depth_label.tex (3 pages)
notes/fig_cut_depth_label.png
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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