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 γt[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 np-complete problem.