, Volume 30, Issue 1, pp 47-69
Date: 17 Nov 2012

Plane Graphs with Parity Constraints

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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.

This research was initiated during the Fifth European Pseudo-Triangulation Research Week in Ratsch an der Weinstraße, Austria, 2008. Research of O. Aichholzer, T. Hackl, A. Pilz, and B. Vogtenhuber was partially supported by the Austrian Science Fund (FWF): S9205-N12, NFN Industrial Geometry. O. Aichholzer and B. Vogtenhuber are partially supported by the ESF EUROCORES programme EuroGIGA-ComPoSe, Austrian Science Fund (FWF): I 648-N18. T. Hackl is funded by the Austrian Science Fund (FWF): P23629-N18. M. Hoffmann is partially supported by the ESF EUROCORES programme EuroGIGA-GraDr, Swiss National Science Foundation (SNF): 20GG21-134306. A. Pilz is recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. G. Rote is partially supported by the ESF EUROCORES programme EuroGIGA-ComPoSe, Deutsche Forschungsgemeinschaft (DFG): FE 340/9-1. Research by B. Speckmann supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707.