The multiple originator broadcasting problem in graphs ?
Given a graph G G and a vertex subset S S of V(G) V ( G ) , the broadcasting time with respect to S S , denoted by b(G,S) b ( G , S ) , is the minimum broadcasting time when using S S as the broadcasting set. And the k k -broadcasting number , denoted by bk(G) b k ( G ) , is defined by bk(G)=min{b(G,S)|S?V(G),|S|=k} b k ( G ) = min { b ( G , S ) | S ? V ( G ) , | S | = k } . Given a graph G G and two vertex subsets S S , S? S ? of V(G) V ( G ) , define d ( v , S ) = min u ? S d ( v , u ) , d(S,S?)=min{d(u,v)|u?S d ( S , S ? ) = min { d ( u , v ) | u ? S , v?S?} v ? S ? } , and d ( G , S ) = max v ? V ( G ) d ( v , S ) for all v?V(G) v ? V ( G ) . For all k k , 1?k?|V(G)| 1 ? k ? | V ( G ) | , the k k -radius  of G G , denoted by rk(G) r k ( G ) , is defined as rk(G)=min{d(G,S)|S?V(G) r k ( G ) = min { d ( G , S ) | S ? V ( G ) , |S|=k} | S | = k } . In this paper, we study the relation between the k k -radius and the k k -broadcasting numbers of graphs. We also give the 2 2 -radius and the 2 2 -broadcasting numbers of the grid graphs, and the k k -broadcasting numbers of the complete n n -partite graphs and the hypercubes.
hypercube k k -radius k k k k k k k -broadcasting number k k k k k cartesian product path