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 $\frac{{\rm poly}(n)}{\sqrt{8}^n} \cdot{\sf pg}(P)$. 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 $\sqrt{8}\approx 2.8284$ 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.

Both authors acknowledge support by SNF project 200021-116741. These results were first presented at the Symposium “Significant Advances in Computer Science” (SACS’07) celebrating 30 years Computer Science at Graz University of Technology, Austria, November 6, 2007.