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

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 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 = Ω(n), where n denotes the number of vertices in D. 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 Email addresses: [email protected] (Laurent Gourvès), [email protected] (Adria Lyra), [email protected] (Carlos A. Martinhon ∗∗ ), [email protected] (Jérôme Monnot) ∗ A preliminary version of this paper appeared in 7th Annual Conference on Theory and Application of Models of Computation, TAMC2010, Czech Republic, [21] ∗∗ Partially supported by FAPERJ/Brazil and CNPq/Brazil Preprint submitted to Discrete Applied Mathematics October 19, 2012 open problem posed in Gutin et. al. [22], whose objective is to determine whether a 2-arc-colored tournament contains a properly arc-colored circuit.