Rewrite Conjecture 1.20 (universal nesting) with the iso notion fixed
to combinatorial with O preserved: rooted tree iso + plane-outerplanar
iso of O on each tread + child/face correspondence, with B_out
explicitly not required to match (essential for sub-tree embedding).
Factor the technical core out as Conjecture 1.22 (seam realizability):
for every k >= 3, exhibit a triangulated planar disk H_k with
boundary a k-cycle whose BFS-from-boundary tree of treads is iso to a
given T_1. Add Remark 1.23 stating that universal nesting reduces to
seam realizability by excise-and-glue using the existing structural
theorems.
Reworked Remark 1.24 (motivation) keeps the compositional-colourability
and universality bullets, and replaces the old open-questions paragraph
with three concrete subproblems: a candidate apex-removal construction
for the seam, 6-connectivity preservation as the relevant 4CT
subproblem, and a justification of why the weaker iso notion is
necessary.
Add fig_seam_construction.png (and the matplotlib script that generates
it) illustrating the seam construction on a 10-vertex G_1 with
T_1 a chain of length 3; the script asserts BFS-from-boundary in H_5
reproduces ell_{G_1} on V(G_1) \ {S_1}, giving a verified small
instance of the conjecture.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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