A overrightarrow{P₃}-decomposition of tournaments and bipartite digraphs

A $\overrightarrow{P_{3}}$-decomposition of a directed graph $D$ is a partition of the arcs of $D$ into directed paths of length $2$. In this paper, we give a characterization for a tournament and a bipartite digraph admitting a $\overrightarrow{P_{3}}$-decomposition. This solves a problem posed by Diwan ($\overrightarrow{P_{3}}$-decomposition of directed graphs, Discrete Appl. Math., http:// dx.doi.org/10.1016/j.dam.2016.01.039.).