Convex polygons in geometric triangulations

TL;DR

This paper establishes a tighter upper bound on the maximum number of convex polygons in triangulations of n points, nearly matching the known lower bound, and provides an efficient method to count convex polygons in planar graphs.

Contribution

It improves the upper bound on convex polygons in triangulations and introduces an efficient counting method for convex polygons in planar graphs.

Findings

Maximum convex polygons in triangulations is O(1.5029^n)

Improved upper bound from previous work

Efficient counting algorithm for convex polygons in planar graphs

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, L\"offler, and Pach (2012) and almost matches the current best lower bound of $\Omega(1.5028^n)$ due to the same authors. Given a planar straight-line graph $G$ with $n$ vertices, we show how to compute efficiently the number of convex polygons in $G$.