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>
Extend the deep embedding to include the outer face, decompose into
quadrilaterals via level-edge pairing on the sphere, and define a
deterministic sequence built from four moves (anchor drop, level add,
join, ring completion) with a recursive lex-smallest tiebreak on the
initial quadrilateral. Attempt the termination theorem and the per-move
case analyses.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis