The previous reading of Def 1.5 had B_in be a simple cycle on V(O).
This forced an artificial (R2) hypothesis on Lemma 1.7 ruling out
multi-hole topology of R_{C'}. But the multi-hole structure is
already captured naturally by the inner outerplanar graph O itself:
when O is not 2-connected (has a bridge, cut-vertex, or multiple
components), its outer-face boundary is a closed walk rather than a
simple cycle, and that walk traverses bridges twice and visits cut-
vertices multiple times.
Changes:
- Definition 1.5: B_in is now defined as the closed walk in O that
traces O's outer-face boundary in the inherited embedding. It is
a simple cycle when O is 2-connected and a non-simple closed walk
in general. B_out remains a simple cycle (or degenerate single
vertex).
- Lemma 1.7: (R2) hypothesis removed. Only (R1) (manifold) remains.
The proof of the boundary structure is rewritten: we identify
B_out as the source-side boundary cycle, O as G[V_{C'} ∩ L_{d+1}]
(outerplanar by Lemma 2.6 of [bauerfeld-pds]), and B_in as O's
outer-face boundary walk. Multi-hole topology of R_{C'} is now
captured by O being non-2-connected or disconnected, without
needing an extra constraint.
- Remark 1.10 (was R1+R2): now Remark 1.10 (R1-when), explaining the
pinch obstruction for (R1) and explicitly noting that multi-hole
topology does NOT require an additional hypothesis under the new
Definition 1.5.
Empirical motivation: in the brute-force search over maximal planar
graphs at n in [7, 11] (30,587 components from all single-vertex
sources), 171 components at depth 1 had R_{C'} with 3 boundary
cycles in the surface-classification sense (n_bdry > 2). Each is a
valid tire graph under the relaxed definition: B_out is the level-d
cycle, O is the level-(d+1) outerplanar subgraph (non-2-connected
when this occurs), and B_in is its outer-face boundary walk.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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