# On Kernels for d-Path Vertex Cover

In this paper we study the kernelization of 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) of length d. It is known that d-PVC is NP-complete for d ≥ 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel of size (2d-1)k^d-1+k. We improve on this by giving better kernels. Specifically, we give O(k^2) size (vertices and edges) kernels for the cases when d = 4 and d = 5. Further, we give an O(k^3) size kernel for d-PVC.

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