We define a set $P$ to be a branching $k$-path vertex cover of an undirected forest $F$ if all leaves and isolated vertices (vertices of degree at most $1$) of $F$ belong to $P$ and every path on $k$ vertices (of length $k-1$) contains either a branching vertex (a vertex of degree at least $3$) or a vertex belonging to $P$. We define the branching $k$-path vertex cover number of an undirected forest $F$, denoted by $ψ_b(F,k)$, to be the number of vertices in the smallest branching $k$-path vertex cover of $F$. These notions for a rooted directed forest are defined similarly, with natural adjustments. We prove the lower bound $ψ_b(F,k) \geq \frac{n+3k-1}{2k}$ for undirected forests, the lower bound $ψ_b(F,k) \geq \frac{n+k}{2k}$ for rooted directed forests, and that both of them are tight.

翻译：我们定义集合$P$为无向森林$F$的一个分支$k$-路径顶点覆盖，如果$F$的所有叶子和孤立顶点（度数至多为1的顶点）都属于$P$，并且每个包含$k$个顶点的路径（长度为$k-1$）要么包含一个分支顶点（度数至少为3的顶点），要么包含一个属于$P$的顶点。我们定义无向森林$F$的分支$k$-路径顶点覆盖数，记作$ψ_b(F,k)$，为$F$的最小分支$k$-路径顶点覆盖中的顶点数。对于有根有向森林，这些概念的定义类似，仅作自然调整。我们证明了对于无向森林的下界$ψ_b(F,k) \geq \frac{n+3k-1}{2k}$，对于有根有向森林的下界$ψ_b(F,k) \geq \frac{n+k}{2k}$，并且两者都是紧的。