The kk-path vertex cover of rooted product graphs

A subset SS of vertices of a graph GG is called a kk-path vertex cover if every path of order kk in GG contains at least one vertex from SS. Denote by ψk(G)ψk(G) the minimum cardinality of a kk-path vertex cover in GG. In this article a lower and an upper bound for ψkψk of the rooted product graphs are presented. Two characterizations are given when those bounds are attained. Moreover ψ2ψ2 and ψ3ψ3 are exactly determined. As a consequence the independence and the dissociation number of the rooted product are given.