Laurent Gourvès ; Adria Lyra ; Carlos A. Martinhon ; Jérôme Monnot - On paths, trails and closed trails in edge-colored graphs

dmtcs:586 - Discrete Mathematics & Theoretical Computer Science, September 3, 2012, Vol. 14 no. 2 -
On paths, trails and closed trails in edge-colored graphsArticle

Authors: Laurent Gourvès 1; Adria Lyra 2; Carlos A. Martinhon 3; Jérôme Monnot ORCID1

  • 1 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
  • 2 Instituto Multidisciplinar [Rio]
  • 3 Instituto de Computação [Niteroi-Rio de Janeiro]

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 problems.

Volume: Vol. 14 no. 2
Section: Graph Theory
Published on: September 3, 2012
Accepted on: June 9, 2015
Submitted on: January 5, 2012
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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