Counting plane graphs: Flippability and its applications

Abstract

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S. We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let denote the maximum number of triangulations on a set of N points in the plane. Then we show [using the known bound tr(N) ≤ 30N] that any N-element point set admits at most 6.9283N · tr(N) ≤ 207.85N crossing-free straight-edge graphs, O(4.7022N) · tr(N) = O(141.07N) spanning trees, and O(5.3514N) · tr(N) = O(160.55N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN, fewer than cN, or more than cN edges, for any constant parameter c, in terms of c and N.