Minimum k-path vertex cover

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 \psi_k(G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining \psi_k(G) is NP-hard for each k \geq 2, while for trees the problem can be solved in linear time. We investigate upper bounds on the value of \psi_k(G) and provide several estimations and exact values of \psi_k(G). We also prove that \psi_3(G) \leq (2n + m)/6, for every graph G with n vertices and m edges.