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 ...

For a graph G and a positive integer k, a subset S of vertices of G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. The cardinality of a minimum k-path vertex cover is denoted by ψk(G). In this paper, we present the exact values of ψk in some product graphs of stars and complete graphs.

Nakano et al. in [20] presented a timeand work-optimal algorithm for finding the smallest number of vertex-disjoint paths that cover the vertices of a cograph and left open the problem of applying their technique into other classes of graphs. Motivated by this issue we generalize their technique and apply it to the class of P4-sparse graphs, which forms a proper superclass of cographs. We show ...