Powers of geometric intersection graphs and dispersion algorithms

Authors:
Geir Agnarsson;Peter Damaschke;Magnús M. Halldórsson
Affiliations:
Computer Science Program, School of Computing, Armstrong Atlantic State University, Savannah, GA;Department of Computer Science, Chalmers University, 41296 Göteborg, Sweden;Department of Computer Science, University of Iceland, IS-107 Reykjavik, Iceland and Iceland Genomics Corporation (UVS), Snorrabraut 60, IS-105 Reykjavik, Iceland
Venue:
Discrete Applied Mathematics - Special issue on stability in graphs and related topics
Year:
2003

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Abstract

We study powers of certain geometric intersection graphs: interval graphs, m-trapezoid graphs and circular-arc graphs. We define the pseudo-product, (G, G') → G*G', of two graphs G and G' on the same set of vertices, and show that G,G' is contained in one of the three classes of graphs mentioned here above, if both G and G' are also in that class and fulfill certain conditions. This gives a new proof of the fact that these classes are closed under taking power; more importantly, we get efficient methods for computing the representation for Gk if k ≥ 1 is an integer and G belongs to one of these classes, with a given representation sorted by endpoints. We then use these results to give efficient algorithms for the k-independent set, dispersion and weighted dispersion problem on these classes of graphs, provided that their geometric representations are given.