Paths and trails in edge-colored graphs

• Published in: Theoretical computer science Vol. 409; no. 3; pp. 497 - 510
• Main Authors: Abouelaoualim, A., Das, K.Ch, Faria, L., Manoussakis, Y., Martinhon, C., Saad, R.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier B.V 28.12.2008; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Computer science; control theory; systems; Connectivity; Edge-colored graphs; Exact sciences and technology; Information retrieval. Graph; Miscellaneous; Properly edge-colored paths; Theoretical computing; Trails and cycles; Properly edge-colored paths; Connectivity; Edge-colored graphs; Trails and cycles; Polynomial; Vertex; Edge(graph); Approximation; Maximization; Computer theory; Color; Algorithmics; Shortest path; Cycle; Graph connectivity
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2008.09.021

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 = Ω ( 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 = Ω ( n ) . We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.