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.

翻译：最近，Campbell、Hörsch 和 Moore 证明，判定一个图是否能边分解为一个匹配和一个 $k$-有界线性森林的问题，对于每个 $k \ge 9$ 是 NP-完全的，而对于 $k=1,2$ 则可在多项式时间内求解。在本文的第一部分，我们通过证明该问题对于每个 $k \ge 3$ 都是 NP-完全的，填补了这一空白。在第二部分，我们证明判定一个图是否能边分解为一个匹配和一个 $k$-有界星形森林的问题对于任意 $k \in \mathbb{N} \cup \{ \infty \}$ 都是多项式可解的，从而回答了 Campbell、Hörsch 和 Moore 在相同论文中提出的另一个问题。