Dominating and irredundant broadcasts in graphs.

A broadcast on a nontrivial connected graph G=(V,E) is a function f:V{0,1,,diam(G)} such that f(v)e(v) (the eccentricity of v) for all vV. The cost of f is (f)=vVf(v). A broadcast f is dominating if each uV is at distance at most f(v) from a vertex v with f(v)1.We use properties of minimal dominating broadcasts to define the concept of an irredundant broadcast on G. We determine conditions under which an irredundant broadcast is maximal irredundant. Denoting the minimum costs of dominating and maximal irredundant broadcasts by b(G) and irb(G) respectively, the definitions imply that irb(G)b(G) for all graphs. We show that b(G)54irb(G) for all graphs G.We also briefly consider the upper broadcast number b(G) and upper irredundant broadcast number IRb(G), and illustrate that the ratio IRb/b is unbounded for general graphs.