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 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-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.
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Abouelaoualim, A., Das, K.C., Faria, L., Manoussakis, Y., Martinhon, C., Saad, R. (2008). Paths and Trails in Edge-Colored Graphs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_62
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DOI: https://doi.org/10.1007/978-3-540-78773-0_62
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