Total [1; k]-dominating sets of some graph products

For two given non-negative integers j and k, a dominating set S of a graph G is a total [ j; k]-set if each vertex of V(G) is dominated by at least j, but not more than k vertices of S . The cardinality of the smallest such set is denoted by t[ j;k](G). In this paper, we investigate the existence of such dominations for corona product of graphs. Moreover, we describe total [1; k]-sets of Cartesian products G H for arbitrary graphs G and H. Also, we give some su cient conditions for G and H such that the strong product G H has a total [1; k]-set.