Counting Plane Graphs: Flippability and Its Applications

  • Michael Hoffmann
  • André Schulz
  • Micha Sharir
  • Adam Sheffer
  • Csaba D. Tóth
  • Emo Welzl


We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S.

We prove a worst-case tight lower bound for the number of pseudo-simultaneously 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 \(\mathsf{tr}(N)\) denote the maximum number of triangulations on a set of N points in the plane. Then we show [using the known bound \(\mathsf{tr}(N) < 3{0}^{N}\)] that any N-element point set admits at most \(6.928{3}^{N} \cdot \mathsf{tr}(N) < 207.8{5}^{N}\) crossing-free straight-edge graphs, \(O(4.702{2}^{N}) \cdot \mathsf{tr}(N) = O(141.0{7}^{N})\) spanning trees, and \(O(5.351{4}^{N}) \cdot \mathsf{tr}(N) = O(160.5{5}^{N})\) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN, fewer than cN, or more than cN edges, for any constant parameter c, in terms of c and N.


Convex Hull Span Tree Planar Graph Convex Polygon Interior Vertex 
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Work on this chapter by Micha Sharir and Adam Sheffer was partially supported by Grant 338/09 from the Israel Science Fund. Work by Micha Sharir was also supported by NSF Grant CCF-08-30272, by Grant 2006/194 from the U.S.-Israel Binational Science Foundation, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by Csaba D. Tóth was supported in part by NSERC Grant RGPIN 35586. Research by this author was conducted at ETH Zürich. Emo Welzl acknowledges support from the EuroCores/EuroGiga/ComPoSe SNF Grant 20GG21_134318/1. Part of the work on this chapter was done at the Centre Interfacultaire Bernoulli (CIB), EPFL, Lausanne, during the Special Semester on Discrete and Computational Geometry, Fall 2010, and was supported by the Swiss National Science Foundation.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Michael Hoffmann
    • 1
  • André Schulz
    • 2
  • Micha Sharir
    • 3
    • 4
  • Adam Sheffer
    • 3
  • Csaba D. Tóth
    • 5
  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  2. 2.Institut für Mathematische Logik und GrundlagenforschungUniversität MünsterMünsterGermany
  3. 3.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  5. 5.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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