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, Orsay Cedex, France;University of Paris-XI, LRI, Orsay Cedex, France;Estadual Univ. of Rio de Janeiro, Dept. of Math., São Gonçalo, RJ, Brazil;University of Paris-XI, LRI, Orsay Cedex, France;Fluminense. Fed. Univ., Instit. of Comput., Niterói, RJ, Brazil;-
Venue:
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Year:
2008

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Abstract

This paper deals with the existence and search of 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 some particular 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 Gc is NP-complete even for k = 2 and c = Ω(n2), where n denotes the number of vertices in Gc. Moreover, we prove that these problems remain NP-complete for c-colored graphs containing no Properly Edge-Colored cycles and c = Ω(n). We obtain some approximation results for those maximization problems together with polynomial results for some particulars classes of edge-colored graphs.