A subset D ⊆ V(G) is called a k-distance dominating set of G if every vertex in V(G)-D is within distance k from some vertex of D. The minimum cardinality among all k-distance dominating sets of G is called the k-distance domination number of G. In this note we give upper bounds on the k-distance domination number of a connected bipartite graph, and improve some results have been given like Theorems 2.1 and 2.7 in [Tian and Xu, A note on distance domination of graphs, Australasian Journal of Combinatorics, 43 (2009), 181-190].
