Article thumbnail
Location of Repository

SOLVING COLORING, MINIMUM CLIQUE COVER AND KERNEL PROBLEMS ON ARC INTERSECTION GRAPHS OF DIRECTED PATHS ON A TREE

By Olivier Durand, De Gevigney, Christian Popa, Julien Reygner and Ayrin Romero

Abstract

Abstract. Let T = (V,A) be a directed tree. Given a collection P of dipaths on T, we can look at the arc-intersection graph I(P, T) whose vertex set is 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 ||P|) (it corresponds to the minimum clique cover of I(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). 1

Year: 2016
OAI identifier: oai:CiteSeerX.psu:10.1.1.963.9849
Provided by: CiteSeerX
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • http://cermics.enpc.fr/%7Emeun... (external link)
  • http://cermics.enpc.fr/%7Emeun... (external link)
  • http://citeseerx.ist.psu.edu/v... (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.