Adds a new definition formalizing the "partial tire facial dual" T'_{f'}:
(i) Annular dual subgraph T'_ann := G'[{d_f : f ∈ F_ann}], with
planar embedding inherited from G' (where G' is the inner
planar dual of the maximal planar G).
(ii) For each face f' of T'_ann in its inherited embedding,
T'_{f'} := closed G'-neighborhood of V(f') together with
every G'-edge incident to V(f').
Adds a remark noting that in the spoke-only case T'_ann = Γ ≅ C_{n+m}
has two faces (both with V(f') = all interior dual vertices), and
T'_{f'} recovers the planar dual of T when G is the tire plus one
source-side and one O-side face.
Paper stays at 9 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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