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 v epsilon V. The cost of f is sigma (f) = Sigma(vev)f (v). A broadcast f is dominating if each u Sigma V 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 gamma(b)(G) and ir(b)(G) respectively, the definitions imply that ir(b)(G) <= gamma(b) (G) for all graphs. We show that gamma(b)(G) <= 5/4 ir(b)(G) for all graphs G.We also briefly consider the upper broadcast number F-b(G) and upper irredundant broadcast number IRb(G), and illustrate that the ratio IRb /F-b is unbounded for general graphs. (C) 2016 Elsevier B.V. All rights reserved.