didericis 47a87a4c2b plane_depth_sequencing: generalise Lemma 2.6 to arbitrary outer-face source; tighten the citation in coloring_nested_tire_graphs ...

In `plane_depth_sequencing/paper.tex`:
- Lemma 2.6 now allows any nonempty source S ⊆ V(G) whose vertices all
  lie on the boundary of the outer face of the chosen embedding,
  rather than only the outer-cycle case S = V(C).
- The proof is the same argument with S in place of C: at d=0 each
  S-vertex remains on the outer face after restriction; for d ≥ 1
  the BFS ball V_{<d}^S is connected and reaches the outer face
  via S.
- The original outer-cycle statement is preserved as a remark inside
  the lemma.
- Adds \label{lem:outerplanarity}.

In `coloring_nested_tire_graphs/paper.tex`:
- The proof of Lemma 1.7 drops the "extends verbatim" caveat and
  simply cites the generalised Lemma 2.6, noting that since the level
  source S is a single vertex (per the local Level-source definition)
  we may freely choose an embedding placing S on the outer face;
  outerplanarity is a graph property so the conclusion transfers.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

2026-05-25 15:24:08 -04:00

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