- Adds a proof sketch to Lemma 1.7 (tire-component lemma). The
outerplanarity step cites Lemma 2.6 of `bauerfeld-pds` (the
Plane Depth Sequencing manuscript), which proves that for any
level source the subgraph induced on a single depth level is
outerplanar. The proof notes that the cited result, originally
stated for an outer-cycle source, extends verbatim to an arbitrary
level source by treating S as the depth-0 set.
- The remainder of the proof: layer containment forces V_{C'} ⊆
L_d ∪ L_{d+1}; a face-by-face boundary analysis shows ∂R_{C'} is
monochromatic in level; connectivity of C' rules out higher genus;
the resulting one or two closed boundary walks give the tire
graph's two boundary parts (with the degenerate case at the BFS
endpoints).
- Adds a thebibliography block with one entry for the cited paper.
The paper grows from 3 to 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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