Counting and Enumerating Pointed Pseudo-triangulations with the Greedy Flip Algorithm

This paper studies (pointed, or minimal) pseudotriangulations for a given point set in the plane. Pseudotriangulations have many properties of triangulations, and have more freedom since polygons with more than three vertices are allowed as long as exactly three have angles less than π. In particular, there is a natural flip operation on every internal edge. We establish fundamental properties of pointed pseudo-triangulations. We also present an algorithm to enumerate the pseudo-triangulations of a given point set, based on the greedy flip of Pocchiola and Vegter. Our two independent implementations agree, and allow us to experimentally verify or disprove conjectures on the numbers of pseudo-triangulations and triangulations of a given point set. (For example, we establish that the number of triangulations is less than the number of pseudo-triangulations for all sets of less than 10 points; the proof for all n is still to be discovered.)