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 >= 2. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with O(dk(d)) edges. We improve on this by giving better kernels. Specifically, we give kernels with O(k(2)) vertices and edges for the cases when d = 4 and d = 5. Further, we give a kernel with O(k(4)d(2d+9)) vertices and edges for general d. (c) 2024 Elsevier Inc. All rights reserved.