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.
This research was initiated during the Fifth European Pseudo-Triangulation Research Week in Ratsch a.d. Weinstraße, Austria, 2008. Research of O. Aichholzer, T. Hackl, and B. Vogtenhuber supported by the FWF [Austrian Fonds zur Förderung der Wissenschaftlichen Forschung] under grant S9205-N12, NFN Industrial Geometry. Research by B. Speckmann supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707.
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Aichholzer, O. et al. (2009). Plane Graphs with Parity Constraints. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_2
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DOI: https://doi.org/10.1007/978-3-642-03367-4_2
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