P3→-decomposition of directed graphs

A P3→-decomposition of a directed graph D is a partition of the arcs of D into directed paths of length 2. We characterize symmetric digraphs that do not admit a P3→-decomposition. We show that the only 2-regular, connected directed graphs that do not admit a P3→-decomposition are obtained from undirected odd cycles by replacing each edge by two oppositely directed arcs. In both cases, we give a linear-time algorithm to find a P3→-decomposition, if it exists.