Pins Π_G at the start to be an embedding placing S on the outer face;
such an embedding exists for any single-vertex source. This collapses
the two-embedding split in the previous proof (one for Lemma 2.6,
another for the topological analysis of R_{C'}) into a single
embedding throughout, and removes the "in either order" ambiguity for
B_out and B_in:
- B_out = G[V_{C'} ∩ L_d]: the boundary of R_{C'} closer to S.
- B_in = G[V_{C'} ∩ L_{d+1}]: the boundary farther from S.
The outerplanarity step now cites Lemma 2.6 of [bauerfeld-pds]
directly (no embedding switch). The "tire structure" step pins the
orientation by S's position on the outer face.
Remark 1.9 (degenerate cases) updated: the orientation ambiguity is
gone, so we state the d=0 case has degenerate B_out and the d=D_max
case has degenerate B_in.
(R1) and (R2) remain — they are graph-theoretic and unaffected by
embedding choice (for 3-connected planar graphs the embedding is
essentially unique by Whitney's theorem, so changing the outer face
cannot untangle pinches or merge multi-hole topology).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
1ac80aa5cf