Counting Plane Graphs: Flippability and Its Applications

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane into so called pseudo-simultaneously flippable edges. We prove a worst-case tight lower bound for the number of pseudosimultaneously 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 tr(N) 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 ċ tr(N) < 207.85N crossing-free straight-edge graphs, O(4.8795N) ċ tr(N) = O(146.39N) spanning trees, and O(5.4723N) ċ tr(N) = O(164.17N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have fewer than cN or more than cN edges, for a constant parameter c, in terms of c and N.

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