On the Path Sequence of a Graph

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 ѱ k(G) the minimum cardinality of a k-path vertex cover in G and form a sequence ѱ(G) = (ѱ(G), ѱ2(G), . . . , ѱV |(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 (G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given.