Enumerating Triangulation Paths

Recently, Aichholzer introduced the remarkable concept of the so-called triangulation path (of a triangulation with respect to a segment), which has the potential of providing efficient counting of triangulations of a point set, and efficient representations of all such triangulations. Experiments support such evidence, although – apart from the basic uniqueness properties – little has been proved so far. In this paper we provide an algorithm which enumerates all triangulation paths (of all triangulations of a given point set with respect to a given segment) in time O(t n3 logn) and O(n) space, where n denotes the number of points and t is the number of triangulation paths. For the algorithm we introduce the notion of flips between such paths, and define a structure on all paths such that the reverse search approach can be applied. We also refute Aichholzer’s conjecture that points in convex position maximize the number of such paths.