Counting Plane Graphs with Exponential Speed-Up

  • Andreas Razen
  • Emo Welzl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6570)


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.


Counting crossing-free configurations plane graphs triangulations constrained Delaunay triangulation edge flips 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andreas Razen
    • 1
  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZurichSwitzerland

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