For some $k \in \mathbb{Z}_{\geq 0}\cup \infty$, we call a linear forest $k$-bounded if each of its components has at most $k$ edges. We will say a $(k,\ell)$-bounded linear forest decomposition of a graph $G$ is a partition of $E(G)$ into the edge sets of two linear forests $F_k,F_\ell$ where $F_k$ is $k$-bounded and $F_\ell$ is $\ell$-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both $k$ and $\ell$ are at least $2$, NP-complete if $k\geq 9$ and $\ell =1$, and is in P for $(k,\ell)=(2,1)$. Before this, the only known NP-complete cases were the $(2,2)$ and $(3,3)$ cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than $3$-edge-colouring such graphs.

翻译：对于某个$k \in \mathbb{Z}_{\geq 0}\cup \infty$，我们称一个线性森林为$k$-有界的，如果其每个分量最多包含$k$条边。我们称图$G$的一个$(k,\ell)$-有界线性森林分解是将$E(G)$划分为两个线性森林$F_k$和$F_\ell$的边集，其中$F_k$是$k$-有界的，$F_\ell$是$\ell$-有界的。我们证明，判定给定图是否存在此类分解的问题在$k$和$\ell$均至少为$2$时是NP完全的，在$k\geq 9$且$\ell=1$时也是NP完全的，而对于$(k,\ell)=(2,1)$的情况属于P类问题。此前已知的NP完全情况仅限于$(2,2)$和$(3,3)$情形。我们的硬度结果回答了Bermond等人于1984年提出的一个问题。我们还证明，围长至少为九的平面图可分解为一个线性森林和一个匹配，这尤其比对此类图进行$3$-边着色更强。