Graphs and Combinatorics

, Volume 30, Issue 1, pp 47–69 | Cite as

Plane Graphs with Parity Constraints

  • Oswin Aichholzer
  • Thomas Hackl
  • Michael Hoffmann
  • Alexander Pilz
  • Günter Rote
  • Bettina Speckmann
  • Birgit Vogtenhuber
Original Paper

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 \({p\in S}\) 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 that satisfy all but at most three parity constraints. For triangulations we can satisfy about 2/3 of the parity constraints and we show that in the worst case there is a linear number of constraints that cannot be fulfilled. In addition, we prove that for a given simple polygon H with polygonal holes on S, it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.

Keywords

Triangulation Vertex degree parity Pseudo-triangulation Geometric graph 

Mathematics Subject Classification (2010)

05C10 52C99 

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Copyright information

© Springer Japan 2012

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 TechnologyGrazAustria
  2. 2.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany
  4. 4.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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