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 $d\ge 2$. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with $O\left(d{k}^d\right)$ edges. We improve on this by giving better kernels. Specifically, we give kernels with $O\left(k^2\right)$ vertices and edges for the cases when $d=4$ and $d=5$. Further, we give a kernel with $O\left(k^4{d}^{2d+9}\right)$ vertices and edges for general d.