Powers of geometric dispersion intersection graphs and algorithms

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 G(k) if k greater than or equal to 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. (C) 2003 Elsevier B.V. All rights reserved.