The multiple originator broadcasting problem in graphs

Abstract

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