On Kernels for d-Path Vertex Cover

In this paper we study the kernelization of the $d$-Path Vertex Cover ($d$-PVC) problem. Given a graph $G$, the problem requires finding whether there exists a set of at most $k$ vertices whose removal from $G$ results in a graph that does not contain a path (not necessarily induced) with $d$ vertices. It is known that $d$-PVC is NP-complete for $d\geq 2$. Since the problem generalizes to $d$-Hitting Set, it is known to admit a kernel with $\mathcal{O}(dk^d)$ edges. We improve on this by giving better kernels. Specifically, we give kernels with $\mathcal{O}(k^2)$ vertices and edges for the cases when $d=4$ and $d=5$. Further, we give a kernel with $\mathcal{O}(k^4d{2d+9})$ vertices and edges for general $d$.