On paths, trails and closed trails in edge-colored graphs

Abstract : In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.
Type de document :
Article dans une revue
Discrete Mathematics and Theoretical Computer Science, DMTCS, 2012, Vol. 14 no. 2 (2), pp.57--74
Liste complète des métadonnées

Littérature citée [25 références]  Voir  Masquer  Télécharger

Contributeur : Service Ist Inria Sophia Antipolis-Méditerranée / I3s <>
Soumis le : mardi 13 mai 2014 - 16:27:39
Dernière modification le : jeudi 11 janvier 2018 - 06:17:30
Document(s) archivé(s) le : lundi 10 avril 2017 - 22:44:50


Fichiers produits par l'(les) auteur(s)


  • HAL Id : hal-00990589, version 1



Laurent Gourvès, Adria Lyra, Carlos A. Martinhon, Jérôme Monnot. On paths, trails and closed trails in edge-colored graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2012, Vol. 14 no. 2 (2), pp.57--74. 〈hal-00990589〉



Consultations de la notice


Téléchargements de fichiers