  A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets
$A_1$ and $A_2$ such that both $(V,A_1)$ and $(V,A_2)$ are strong. Let $T$ be a
digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs
such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the
composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set
$\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set
$$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le
j_i\le n_i, 1\le q_p\le n_p\}.$$
  For digraph compositions $Q=T[H_1,\dots H_t]$, we obtain sufficient
conditions for $Q$ to have a good decomposition and a characterization of $Q$
with a good decomposition when $T$ is a strong semicomplete digraph and each
$H_i$ is an arbitrary digraph with at least two vertices.
  For digraph products, we prove the following: (a) if $k\geq 2$ is an integer
and $G$ is a strong digraph which has a collection of arc-disjoint cycles
covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the
$k$th powers with respect to Cartesian product) has a good decomposition; (b)
for any strong digraphs $G, H$, the strong product $G\boxtimes H$ has a good
decomposition.
