Paths and trails in edge-colored graphs

Authors:
A. Abouelaoualim;K. Ch. Das;L. Faria;Y. Manoussakis;C. Martinhon;R. Saad
Affiliations:
University of Paris-XI, LRI, Bât. 490, 91405 Orsay Cedex, France;University of Paris-XI, LRI, Bât. 490, 91405 Orsay Cedex, France;Estadual University of Rio de Janeiro, Department of Math., São Gonçalo, RJ, Brazil;University of Paris-XI, LRI, Bât. 490, 91405 Orsay Cedex, France;University of Paris-XI, LRI, Bât. 490, 91405 Orsay Cedex, France and Fluminense Federal University, Institute of Computation, Niterói, RJ, 24210-240, Brazil;114/40 rue Charles Albanel, Gatineau (QC) J8Z 1R2, Canada
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s-t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s and t for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint properly edge-colored s-t paths/trails in a c-edge-colored graph G^c is NP-complete even for k=2 and c=@W(n^2), where n denotes the number of vertices in G^c. Moreover, we prove that these problems remain NP-complete for c-edge-colored graphs containing no properly edge-colored cycles and c=@W(n). We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.