Some Results on [1, k]-sets of Lexicographic Products 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 γt[1,k](G). We consider [1, k]-sets that are also independent. Note that not every graph has an independent [1, k]-set. For graphs having an independent [1, k]-set, we define [1, k]-independence numbers which is denoted by γi[1,k](G). In this paper, we investigate the existence of [1, k]-sets in lexicographic products G ◦H. Furthermore, we completely characterize graphs which their lexicographic product has at least one total [1, k]set. Also, we determine γ[1,k](G ◦H), γt[1,k](G ◦H) and γi[1,k](G ◦H). Finally, we show that finding smallest total [1, k]-set is NP -complete.