The multiple originator broadcasting problem in graphs

Given a graph GG and a vertex subset SS of V(G)V(G), the broadcasting time with respect to  SS, denoted by b(G,S)b(G,S), is the minimum broadcasting time when using SS as the broadcasting set. And the kk-broadcasting number  , denoted by bk(G)bk(G), is defined by bk(G)=min{b(G,S)|S⊆V(G),|S|=k}bk(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph GG and two vertex subsets SS, S′S′ of V(G)V(G), define d(v,S)=minu∈Sd(v,u), d(S,S′)=min{d(u,v)|u∈Sd(S,S′)=min{d(u,v)|u∈S, v∈S′}v∈S′}, and d(G,S)=maxv∈V(G)d(v,S) for all v∈V(G)v∈V(G). For all kk, 1⩽k⩽|V(G)|1⩽k⩽|V(G)|, the kk-radius   of GG, denoted by rk(G)rk(G), is defined as rk(G)=min{d(G,S)|S⊆V(G)rk(G)=min{d(G,S)|S⊆V(G), |S|=k}|S|=k}.In this paper, we study the relation between the kk-radius and the kk-broadcasting numbers of graphs. We also give the 22-radius and the 22-broadcasting numbers of the grid graphs, and the kk-broadcasting numbers of the complete nn-partite graphs and the hypercubes.