How Many Potatoes are in a Mesh?

TL;DR

This paper investigates the maximum number of convex polygons that can be formed from triangulations of n vertices, providing bounds that depend on the triangulation's geometric properties.

Contribution

It establishes exponential bounds for general triangulations and polynomial bounds for fat and compact triangulations, advancing understanding of polygon counts in geometric graphs.

Findings

Exponential lower bound of Omega(1.5028^n) for general triangulations.

Polynomial upper bound of O(1.62^n) in the worst case.

Polynomial bounds for fat and compact triangulations, with Theta(n^2) and Theta(n) for convex outerplanar polygons.

Abstract

We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has Omega(1.5028^n) convex polygons, and prove an O(1.62^n) upper bound in the worst case. If the triangulation is fat (every triangle has its angles lower-bounded by a constant delta>0), then there can be only polynomially many. We also consider the problem of counting convex outerplanar polygons (i.e., they contain no vertices of the triangulation in their interiors) in the same triangulations. In this setting, we get the same exponential bounds in general triangulations, and lower polynomial bounds in fat triangulations. If the triangulation is furthermore compact (the ratio between the longest and shortest distance between any two vertices is bounded), the bounds drop further to Theta (n^2) for general convex outerplanar polygons, and Theta (n) for fat convex outerplanar polygons.