Let Gbe a simple graph. A subset Sof  is a -dominating set of Gif for each  there exist distinct vertices  such that  and  The minimum cardinality of a -dominating set of Gis called the -domination number of G. In this paper, we characterize the -dominating sets in the join, corona, lexicographic product, and the Cartesian product of two graphs. As direct consequences, we determine the -domination number of each of these graphs.

domination, (1,2)-domination, join, corona, lexicographic product, Cartesian product.