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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aichholzer, O., Hackl, T., Huemer, C., Hurtado, F., Vogtenhuber, B.: Large bichromatic point sets admit empty monochromatic 4-gons (submitted) (2008)
Aichholzer, O., Krasser, H.: The point set order type data base: A collection of applications and results. In: Proc. 13th Canadian Conference on Computational Geometry, Waterloo, Ontario, Canada, pp. 17–20 (2001)
Bose, P.: On embedding an outer-planar graph in a point set. Computational Geometry: Theory and Applications 23(3), 303–312 (2002)
Bose, P., McAllister, M., Snoeyink, J.: Optimal algorithms to embed trees in a point set. Journal of Graph Algorithms and Applications 1(2), 1–15 (1997)
Colbourn, C., Booth, K.: Linear time automorphism algorithms for trees, interval graphs, and planar graphs. SIAM Journal on Computing 10(1), 203–225 (1981)
Erdös, P., Gallai, T.: Graphs with prescribed degree of vertices. Mat. Lapok 11, 264–274 (1960)
Jansen, K.: One strike against the min-max degree triangulation problem. Computational Geometry: Theory and Applications 3(2), 107–120 (1993)
Kettner, L., Kirkpatrick, D., Mantler, A., Snoeyink, J., Speckmann, B., Takeuchi, F.: Tight degree bounds for pseudo-triangulations of points. Computational Geometry: Theory and Applications 25(1&2), 1–12 (2003)
Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11(2), 329–343 (1982)
Pilz, A.: Parity properties of geometric graphs. Master’s thesis, Graz University of Technology, Austria (in preparation, 2009)
Tamura, A., Tamura, Y.: Degree constrained tree embedding into points in the plane. Information Processing Letters 44, 211–214 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-03367-4_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03366-7
Online ISBN: 978-3-642-03367-4
eBook Packages: Computer ScienceComputer Science (R0)