Girth of directed graphs

The definition of girth of an undirected graph is defined as the length of the smallest cycle in the graph. Some directed graphs have no cycle (a directed path that stars and ends at the same vertex) but has two different directed paths between two vertices $v$ and $w$, say both paths go from $v$ to $w$. For a Cayley hash function that corresponds to a collision so it is desirable that the Cayley graph associated to the hash function to have large girth. How one can define the girth in a directed graph?

The girth for directed graphs is usually defined as the directed girth, that is, the minimum length of a directed cycle (or $\infty$ if no directed cycles exist). For instance, see