LAUSR

Let $$G=(V,E)$$G=(V,E) be a graph. A set $$S\subseteq V$$S⊆V is a total k-dominating set if every vertex $$v\in V$$v∈V has at least k neighbors in S. The total k-domination number $$\gamma _{kt}(G)$$γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of the Cartesian product of two graphs, and we investigate the relationship between the total k-domination number of the Cartesian product graph with respect to the total k-domination number in the factors of the product. We also study the total k-domination number in certain particular cases of Cartesian products of graphs and determine the exact values of this parameter.