Abstract: Given a graph and a vertex subset of , the broadcasting time with respect to , denoted by , is the minimum broadcasting time when using as the broadcasting set. And the -broadcasting number, denoted by , is defined by . Given a graph and two vertex subsets , of , define , , , and for all . For all , , the -radius of , denoted by , is defined as , . In this paper, we study the relation between the -radius and the -broadcasting numbers of graphs. We also give the -radius and the -broadcasting numbers of the grid graphs, and the -broadcasting numbers of the complete -partite graphs and the hypercubes. [Copyright &y& Elsevier]