  We consider two decomposition problems in directed graphs. We say that a
digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its
connected components contains at most $k$ arcs.
  For the first problem, a directed linear forest is a collection of
vertex-disjoint directed paths and we consider the problem of decomposing a
given digraph into a $k$-bounded and an $\ell$-bounded directed linear forest
for some fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. We give a full
dichotomy for this problem by showing that it can be solved in polynomial time
if $k+\ell \leq 3$ and is NP-complete otherwise. This answers a question of
Campbell, H\"orsch, and Moore.
  For the second problem, we say that an out-galaxy is a vertex-disjoint
collection of out-stars. Again, we give a full dichotomy of when a given
digraph can be edge-decomposed into a $k$-bounded and an $\ell$-bounded
out-galaxy for fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. More
precisely, we show that the problem can be solved in polynomial time if
$\min\{k,\ell\}\in \{1,\infty\}$ and is NP-complete otherwise.
