Complexity of Paths, Trails and Circuits in Arc-Colored Digraphs

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 D-c 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 = Omega(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, even if c = 2. As a consequence, we solve a weak version of an open problem posed in Gutin et. al. [17].