  Deciding whether a graph can be edge-decomposed into a matching and a
$k$-bounded linear forest was recently shown by Campbell, H{\"o}rsch and Moore
to be NP-complete for every $k \ge 9$, and solvable in polynomial time for
$k=1,2$. In the first part of this paper, we close this gap by showing that
this problem is in NP-complete for every $k \ge 3$. In the second part of the
paper, we show that deciding whether a graph can be edge-decomposed into a
matching and a $k$-bounded star forest is polynomially solvable for any $k \in
\mathbb{N} \cup \{ \infty \}$, answering another question by Campbell,
H{\"o}rsch and Moore from the same paper.
