On the k-path vertex cover of some graph products

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for ψk of the Cartesian and the direct product of paths are derived. It is shown that for ψ3 those bounds are tight. For the lexicographic product bounds are presented for ψk, moreover ψ2 and ψ3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.