Adds Definition 1.7 (Partial tire dual) formalising the user's
construction: for a tire graph T with annular face set F_{ann}, the
partial tire dual D(T) has
- Interior vertices d_f for each annular face f,
- Leaf vertices for each edge of B_out and each occurrence of an
edge on the boundary walk B_in (so cut-vertices/bridges of O
contribute multiple leaves),
- Interior dual edges for each annular edge incident to two annular
faces,
- Leaf edges from d_f to the corresponding leaf for each boundary
edge of the annular region.
Adds Proposition 1.8 showing that when the annular triangulation is
spoke-only (i.e. every annular edge has one endpoint on B_out and one
on B_in) and O is 2-connected, each annular face has exactly 1
boundary edge + 2 interior annular edges. Consequently each interior
vertex d_f has degree 3 = 2 (cycle) + 1 (leaf), and the induced
subgraph on {d_f} is a single cycle of length n + m. D(T) is then
isomorphic to the corona C_{n+m} ∘ K_1 -- a cycle of length n+m with
one leaf attached to each cycle vertex; |V(D(T))| = |E(D(T))| = 2(n+m).
Subsequent numbering shifted: Proposition (Source-side simple-cycle
property) is now 1.9; Lemma (Tire-component) is now 1.10; Remarks
shift to 1.11 and 1.12. All cross-references are by label, so they
update automatically.
Paper grows from 6 to 7 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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