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. Such edges are related to the notion of convex decompositions spanned by S. We derive a worst-case tight lower bound on the number of pseudo simultaneouslyflippable edges in any triangulation, and use this bound to obtain upper bounds for the maximal number of several types of plane graphs that can be embedded (with crossing-free straight edges) on a fixed set of N points in the plane. More specifically, denoting by tr(N) < 30N the maximum possible number of triangulations of a set of N points in the plane, we show that every set of N points in the plane can have at most 6.9283N

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Michael Hoffmann, Micha Sharir, Adam Sheffer, Csaba D. Tóth, Emo Welzl
• Keywords: CORR 2010, Education, Plane Graphs, Simultaneously-flippable Edges, Upper Bounds