Convex Polygons in Geometric Triangulations

Abstract

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029 n ). This improves an earlier bound of O(1.6181 n ) established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of Ω(1.5028 n ) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

Monotone Paths in Geometric Triangulations

“…The number of geometric configurations contained (as a subgraph) in a triangulation of n points have been considered only recently. The maximum number of convex polygons is known to be between Ω(1.5028 n ) and O(1.5029 n ) [9,16]. For the number of monotone paths, Dumitrescu et al [5] gave an upper bound of O(1.8393 n ); we briefly review their proof in Section 2.…”

Counting Carambolas

“…They constructed n-vertex triangulations containing Ω(1.5028 n ) convex polygons, and proved that every triangulation on n points in the plane contains O(1.6181 n ) convex polygons. Dumitrescu and Tóth [9] subsequently sharpened the upper bound to O(1.5029 n ), thereby almost closing the gap between the upper and lower bounds.…”