A subset S of vertices of a graph G = (V,E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by Ïˆk(G) the minimum cardinality of a k-path vertex cover in G and form a sequence Ïˆ(G) = (Ïˆ1(G), Ïˆ2(G), . . . , Ïˆ|V |(G)), called the path sequence of G. In this paper we prove necessary and sufficientâ€¦ (More)