The previous bridge example used a pendant edge as the bridge. A
pendant edge IS technically a bridge (single-edge cut), but the
intended notion was a "proper" non-pendant bridge: an edge cut
connecting two non-trivial subgraphs. Replaced with the smallest
example:
- B_out = 4-cycle on {0, 1, 2, 3}.
- O = barbell on {4..9}: two disjoint triangles {4, 5, 6} and
{7, 8, 9} connected by the bridge edge 6-7.
- Annular triangulation by hand (12 triangles, all listed in the
generator script with planarity verified by Sage).
The barbell case is structurally cleaner: BOTH endpoints of the
bridge have degree >= 2 in O, and the interior dual subgraph has
the two bridge-incident annular faces (d_5, d_6) as its trivalent
theta-graph vertices (in the pendant case, the trivalent vertices
were NOT bridge-incident, which was confusing).
Updates fig_partial_tire_dual_bridge.png and the figure caption.
Paper stays at 9 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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