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On paths, trails and closed trails in edge-colored graphs (2012)
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BibTeX
@MISC{Gourvès12onpaths,,
author = {Laurent Gourvès and Adria Lyra and Carlos A. Martinhon and Jérôme Monnot},
title = {On paths, trails and closed trails in edge-colored graphs},
year = {2012}
}
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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
Keyphrases
polynomial time edge-colored graph pec cycle pec s-t path optimal solution pec path trail subgraph different color vertex disjoint pec s-t path maximum path disjoint pec s-t path monochromatic vertex pec s-t trail c-edge-colored graph vertex disjoint s-t path strong sense algorithmic perspective disjoint pec path approximation algorithm different question maximum vertex arbitrary c-edge-colored graph s-t path related problem s-restricted vertex
