Minimum k-path vertex cover

International audienceA 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 P_k(G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining P_k(G) is NP-hard for each k ≥ 2, while for trees the problem can be solved in linear time. We investigate upper bounds on the value of P_k(G) and provide several estimations and exact values of P_k(G). We also prove that P_3(G) ≤ (2n + m)/6, for every graph G with n vertices and m edges