Complexity of trails, paths and circuits in arc-colored digraphs

  • Authors:

  • Laurent GourvèS;Adria Lyra;Carlos A. Martinhon;JéRôMe Monnot

  • Affiliations:

  • Université Paris-Dauphine, LAMSADE, F-75775, Paris, France;Universidade Federal Rural do Rio de Janeiro, UFRRF, Instituto Multidisciplinar, Brazil;Fluminense Federal University, Institute of Computation, Niterói, RJ, Brazil;Université Paris-Dauphine, LAMSADE, F-75775, Paris, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

We deal with different algorithmic questions regarding properly arc-colored s-t trails, paths and circuits in arc-colored digraphs. Given an arc-colored digraph D^c with c=2 colors, we show that the problem of determining the maximum number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of a properly arc-colored s-t path is NP-complete even for planar digraphs containing no properly arc-colored circuits and c=@W(n), where n denotes the number of vertices in D^c. If the digraph is an arc-colored tournament, we show that deciding whether it contains a properly arc-colored circuit passing through a given vertex x (resp., properly arc-colored Hamiltonian s-t path) is NP-complete for c=2. As a consequence, we solve a weak version of an open problem posed in Gutin et al. (1998) [23], whose objective is to determine whether a 2-arc-colored tournament contains a properly arc-colored circuit.