On path sequences of graphs

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 ψk(G) the minimum cardinality of a k-path vertex cover in G and form a sequence ψ(G) = (ψ1(G), ψ2(G), . . . , ψ|V |(G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in ψ(G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given.