Counting Plane Graphs: Flippability and Its Applications

  • Michael Hoffmann
  • Micha Sharir
  • Adam Sheffer
  • Csaba D. Tóth
  • Emo Welzl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)

Abstract

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane into so called pseudo-simultaneously flippable edges.

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 tr(N) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N) < 30N) that any N-element point set admits at most 6.9283N ·tr(N) < 207.85N crossing-free straight-edge graphs, O(4.8795N) ·tr(N) = O(146.39N) spanning trees, and O(5.4723N) ·tr(N) = O(164.17N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have fewer than cN or more than cN edges, for a constant parameter c, in terms of c and N.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael Hoffmann
    • 1
  • Micha Sharir
    • 2
    • 3
  • Adam Sheffer
    • 2
  • Csaba D. Tóth
    • 4
  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  2. 2.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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