Graphs and Combinatorics

, Volume 23, Supplement 1, pp 67–84 | Cite as

On the Number of Plane Geometric Graphs

  • Oswin Aichholzer
  • Thomas Hackl
  • Clemens Huemer
  • Ferran Hurtado
  • Hannes Krasser
  • Birgit Vogtenhuber
Article

Abstract

We investigate the number of plane geometric, i.e., straight-line, graphs, a set S of n points in the plane admits. We show that the number of plane geometric graphs and connected plane geometric graphs as well as the number of cycle-free plane geometric graphs is minimized when S is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide a unified proof that the cardinality of any family of acyclic graphs (for example spanning trees, forests, perfect matchings, spanning paths, and more) is minimized for point sets in convex position.

In addition we construct a new maximizing configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*\({{(\sqrt{72}\,}^n)}\) = Θ*(8.4853n) triangulations (omitting polynomial factors), improving the previously known best maximizing examples.

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Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Thomas Hackl
    • 1
  • Clemens Huemer
    • 2
  • Ferran Hurtado
    • 2
  • Hannes Krasser
    • 3
  • Birgit Vogtenhuber
    • 1
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.Institute for Theoretical Computer ScienceGraz University of TechnologyGrazAustria

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