Broadcasts in graphs

We say that a function f:V→{0,1,…,diam(G)}f:V→{0,1,…,diam(G)} is a broadcast   if for every vertex v∈Vv∈V, f(v)⩽e(v)f(v)⩽e(v), where diam(G)diam(G) denotes the diameter of G   and e(v)e(v) denotes the eccentricity of vv. The cost   of a broadcast is the value f(V)=∑v∈Vf(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.