Powers of geometric intersection graphs and dispersion algorithms

Paper in proceeding, 2002

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 ≥ 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.