l-path vertex cover is easier than l-hitting set for small l

In the $l$-path vertex cover problem the input is an undirected graph $G$ and an integer $k$. The goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that $G-S$ does not contain a path with $l$ vertices. In this paper we give parameterized algorithms for $l$-path vertex cover for $l = 5,6,7$, whose time complexities are $O^*(3.945k)$, $O^*(4.947k)$, and $O^*(5.951k)$, respectively.