On Kernels for d-Path Vertex Cover

Radovan Červený, 20 Sep 2021

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 \(\mathsf{NP}\)-complete for \(d\geq 2\). Since the problem generalizes to \(d\)-Hitting Set, it is known to admit a kernel of size \((2d−1)kd−1+k\). We improve on this by giving better kernels. Specifically, we give \(\mathcal{O}(k^2)\) size (vertices and edges) kernels for the cases when \(d=4\) and \(d=5\). Further, we give an \(\mathcal{O}(k^3)\) size kernel for \(d\)-PVC.