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Counting Plane Graphs: Flippability and Its Applications

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Thirty Essays on Geometric Graph Theory

Abstract

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.

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Notes

  1. 1.

    Up-to-date bounds for these and for other families of graphs can be found in http://www.cs.tau.ac.il/~sheffera/counting/PlaneGraphs.html (version of November 2010).

  2. 2.

    In the notations O  ∗ (), \({\Theta }^{{_\ast}}()\), and \({\Omega }^{{_\ast}}()\), we neglect polynomial factors.

  3. 3.

    Here we implicitly assume that N is even. The case where N is odd is handled in the exact same manner, since a constant change in the size of F does not affect the asymptotic bounds.

  4. 4.

    This is not quite correct: When j is close to N ∕ 2, the former bound is smaller [e.g., it is O  ∗ (5N) for j = N ∕ 2], but we do not know how to exploit this observation to improve the bound.

  5. 5.

    We need to construct a quadrangulation of the annulus-like region between Q and the convex hull of S. We start by connecting a vertex of Q to a vertex of the convex hull, and in each step we add a quadrangle by either marching along two edges of the hull or along one edge of the hull and one edge of Q. This produces the desired quadrangulation.

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Acknowledgements

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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Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zürich, 8092, Zürich, Switzerland

    Michael Hoffmann & Emo Welzl

  2. Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Münster, Germany

    André Schulz

  3. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Micha Sharir & Adam Sheffer

  4. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Micha Sharir

  5. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

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  1. Michael Hoffmann
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Correspondence to Adam Sheffer .

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  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

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Hoffmann, M., Schulz, A., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E. (2013). Counting Plane Graphs: Flippability and Its Applications. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_16

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