Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

Abstract

Let T = (V, A) be a directed tree. Given a collection \({\mathcal{P}}\) of dipaths on T, we can look at the arc-intersection graph \({I(\mathcal{P},T)}\) whose vertex set is \({\mathcal{P}}\) and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) and proved that they are perfect. DE graphs find one of their applications in the context of optical networks. For instance, assigning wavelengths to set of dipaths in a directed tree network consists in finding a proper coloring of the arc-intersection graph. In the present paper, we give a simple algorithm finding a minimum proper coloring of the paths. a faster algorithm than previously known ones finding a minimum multicut on a directed tree. It runs in \({O(|V||\mathcal{P}|)}\) (it corresponds to the minimum clique cover of \({I(\mathcal{P},T)}\)). a polynomial algorithm computing a kernel in any DE graph whose edges are oriented in a clique-acyclic way. Even if we know by a theorem of Boros and Gurvich that such a kernel exists for any perfect graph, it is in general not known whether there is a polynomial algorithm (polynomial algorithms computing kernels are known only for few classes of perfect graphs).