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

  • Authors:

  • Laurent Gourvès;Adria Lyra;Carlos Martinhon;Jérôme Monnot

  • Affiliations:

  • ,CNRS, FRE 3234, Paris, France;,Inst of Comp., Fluminense Federal University, Niterói, RJ, Brazil;Inst of Comp., Fluminense Federal University, Niterói, RJ, Brazil;,CNRS, FRE 3234, Paris, France

  • Venue:
  • TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
  • Year:
  • 2010

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Abstract

We deal with different algorithmic questions regarding properly arc-colored s-t paths, trails and circuits in arc-colored digraphs Given an arc-colored digraph Dc with c≥2 colors, we show that the problem of maximizing the number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time Surprisingly, we prove that the determination of one properly arc-colored s-t path is NP-complete even for planar digraphs containing no properly arc-colored circuits and c=Ω(n), where n denotes the number of vertices in Dc 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, even if c=2 As a consequence, we solve a weak version of an open problem posed in Gutin et al. [17].