Counting plane graphs with exponential speed-up

Abstract

We show that one can count the number of crossing-free geometric graphs on a given planar point set exponentially faster than enumerating them. More precisely, given a set P of n points in general position in the plane, we can compute pg(P), the number of crossing-free graphs on P, in time at most poly(n)/√8n. No similar statements are known for other graph classes like triangulations, spanning trees or perfect matchings. The exponential speed-up is obtained by enumerating the set of all triangulations and then counting subgraphs in the triangulations without repetition. For a set P of n points with triangular convex hull we further improve the base √8≈2.8284 of the exponential to 3.347. As a main ingredient for that we show that there is a constant α>0 such that a triangulation on P, drawn uniformly at random from all triangulations on P, contains, in expectation, at least n/α non-flippable edges. The best value for α we obtain is 37/18. © 2011 Springer-Verlag Berlin Heidelberg.