Two TikZ figures added to the outerplanarity theorem:
Figure (Case 1, disk tread): apex v_0 at center, hexagonal
non-degenerate boundary (red), 6 spokes (grey) forming a fan of
6 triangles. Dual Γ (blue) is the cycle C_6 connecting the 6
triangle centroids. Outerplanar trivially.
Figure (Case 2, annulus tread): two concentric hexagons for
B_out and B_in, spokes + one extra "bridge-style" interior
annular edge. Dual Γ is a Hamilton cycle of length 12 around the
annulus, plus one chord (dashed). All vertices on outer face →
outerplanar.
Also corrected the Case 1 proof: the disk has a single interior
vertex (the apex), so the triangulation is a FAN around the apex
(not a polygon-triangulation with no interior vertices), and Γ
is a cycle of length k (not a tree). This is still outerplanar.
Added tikz + backgrounds library to preamble.
Page count: 8 → 9.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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