How Many Potatoes are in a Mesh?

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.