Let  be a simple graph. A subset S of  is a -dominating set of G if it is a dominating set of G such that for each  there exists  such that  The smallest cardinality of a -dominating set of G, denoted by  is called the -domination number of G. In this paper, we characterize the -dominating sets of the join, the corona, and the lexicographic product of graphs and determine bounds for the -domination numbers of some of these graphs.

domination, -domination, join, corona, lexicographic product.