Original Paper

Graphs and Combinatorics

, Volume 30, Issue 1, pp 47-69

Plane Graphs with Parity Constraints

  • Oswin AichholzerAffiliated withInstitute for Software Technology, Graz University of Technology
  • , Thomas HacklAffiliated withInstitute for Software Technology, Graz University of Technology
  • , Michael HoffmannAffiliated withInstitute of Theoretical Computer Science, ETH Zürich
  • , Alexander PilzAffiliated withInstitute for Software Technology, Graz University of Technology Email author 
  • , Günter RoteAffiliated withInstitut für Informatik, Freie Universität Berlin
  • , Bettina SpeckmannAffiliated withDepartment of Mathematics and Computer Science, TU Eindhoven
  • , Birgit VogtenhuberAffiliated withInstitute for Software Technology, Graz University of Technology

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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