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Counting Plane Graphs: Flippability and Its Applications

In: Thirty Essays on Geometric Graph Theory

Author

Listed:
  • Michael Hoffmann

    (ETH Zürich, Institute of Theoretical Computer Science)

  • André Schulz

    (Universität Münster, Institut für Mathematische Logik und Grundlagenforschung)

  • Micha Sharir

    (Tel Aviv University, School of Computer Science
    New York University, Courant Institute of Mathematical Sciences)

  • Adam Sheffer

    (Tel Aviv University, School of Computer Science)

  • Csaba D. Tóth

    (University of Calgary, Department of Mathematics and Statistics)

  • Emo Welzl

    (ETH Zürich, Institute of Theoretical Computer Science)

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)

Suggested Citation

  • Michael Hoffmann & André Schulz & Micha Sharir & Adam Sheffer & Csaba D. Tóth & Emo Welzl, 2013. "Counting Plane Graphs: Flippability and Its Applications," Springer Books, in: János Pach (ed.), Thirty Essays on Geometric Graph Theory, pages 303-325, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-0110-0_16
    DOI: 10.1007/978-1-4614-0110-0_16
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