The Number of Crossing Free Configurations on Finite Point Sets in the Plane

Abstract

We consider the family of crossing-free geometric graphs of a certain type—most notably triangulations, but also spanning (Hamiltonian) cycles, spanning trees, matchings, etc.—on a given point set in the plane. In particular, we address the question of how large these families can be in terms of the number of points. After the issue was raised for Hamiltonian cycles by Newborn and Moser, and for triangulations by Avis, it was shown in 1982 by Ajtai, Chvátal, Newborn, and Szemerédi that for any set of n points the number of all crossing-free geometric graphs on is at most c ⁿ for c=10¹³ (as opposed to the previously known bounds of the form c ^{n logn}). We report on some of the developments since then, e.g. a 43ⁿ bound on the number of triangulations whose proof takes a detour via random triangulations.

While this problem seems elusive despite of some progress, related algorithmic questions are even less understood: For example the complexity of determining or approximating the number of triangulations of a point set, or generating a triangulation uniformly at random from all triangulations of a point set.