Plane Graphs with Parity Constraints

  • Oswin Aichholzer
  • Thomas Hackl
  • Michael Hoffmann
  • Alexander Pilz
  • Günter Rote
  • Bettina Speckmann
  • Birgit Vogtenhuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5664)

Abstract

Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p ∈ S, if the parity of the degree of p in G matches its label. In this paper we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation which satisfy all but at most three parity constraints. With triangulations we can satisfy about 2/3 of all parity constraints. In contrast, for a given simple polygon H with polygonal holes on S, we show that it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.

Keywords

Parity Constraint Planar Graph Hamiltonian Cycle Degree Sequence Simple Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Thomas Hackl
    • 1
  • Michael Hoffmann
    • 2
  • Alexander Pilz
    • 1
  • Günter Rote
    • 3
  • Bettina Speckmann
    • 4
  • Birgit Vogtenhuber
    • 1
  1. 1.Institute for Software TechnologyGraz University of TechnologyAustria
  2. 2.Institute for Theoretical Computer ScienceETH ZürichSwitzerland
  3. 3.Institut für InformatikFU BerlinGermany
  4. 4.Dep. of Mathematics and Computer ScienceTU EindhovenThe Netherlands

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