Total [1, k]-Sets in the Lexicographic Product of Graphs

A subset S ⊆ V in a graph G = (V,E) is called a[1, k]-set, if for every vertex v ∈ V S, 1 ≥ |NG(v) ∩S| ≥k. The [1, k]-domination number of G, denoted by γ[1,k](G) is the size of the smallest[1, k]-sets of G. A set S ≥⊆ V (G) is called a total [1, k]-set, if for every vertex v ∈ V , 1 ≥|NG(v) ∩ S| ≥k. If a graph G has at least one total [1, k]-set then the cardinality of the smallest such set is denoted by yt[1,k](G). In this paper ,we investigate the existence of[1, k]-sets in lexicographic products G ◦ H. Furthermore, we completely characterize graphs whose lexicographic product has at least one total[1, k]-set. Finally, we show that finding smallest total[1, k]-set is an N P –complete problem.