LAUSR

A subset D of the vertex set V ( G ) of a graph G is called a [1,  k ]-dominating set if every vertex from $$V-D$$ V - D is adjacent to at least one vertex and at most k vertices of D . A [1,  k ]-dominating set with minimum number of vertices is called a $$\gamma _{[1,k]}(G)$$ γ [ 1 , k ] ( G ) -set, and the number of its vertices is called the [1,  k ]-domination number of G and is denoted by $$\gamma _{[1,k]}(G)$$ γ [ 1 , k ] ( G ) . In this paper, we express the computation of the [1,  k ]-domination number of lexicographic products $$G\circ H$$ G ∘ H as an optimization problem over certain partitions of V ( G ). Nonetheless, in special cases explicit formulas are possible.