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 ∈ V . The cost of f is σWe 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 γ b (G) and ir b (G) respectively, the definitions imply that ir b (G) ≤ γ b (G) for all graphs. We show that γ b (G) ≤ 5 4 ir b (G) for all graphs G. We also briefly consider the upper broadcast number Γ b (G) and upper irredundant broadcast number IR b (G), and illustrate that the ratio IR b /Γ b is unbounded for general graphs. Hedetniemi and Miller [6] introduced the concept of irredundance as precisely the property that makes a dominating set minimal dominating.Ahmadi, Fricke, Schroeder, Hedetniemi and Laskar [1] use a property that makes a dominating broadcast minimal dominating, which was first mentioned in [10], to define broadcast irredundance, which we state here in Section 2.4. The broadcast irredundance number of G is defined asThe definitions imply that ir b (G) ≤ γ b (G) for all graphs G, and as our main result we prove that the ratio γ b / ir b is bounded:After defining our basic concepts in Section 2, we present some properties of irredundant broadcasts in Section 3, the most important of which is a necessary and sufficient condition for an irredundant broadcast to be maximal irredundant (Theorem 7). Theorem 1 is proved in Section 4. We briefly discuss upper broadcast domination and irredundance in Section 5, illustrating that the ratio IR b /Γ b is unbounded for general graphs, and conclude with a list of open problems and conjectures in Section 6. DefinitionsThis section contains more definitions concerning dominating broadcasts, neighbourhoods and boundaries of broadcasting vertices, minimal dominating broadcasts and, finally, irredundant broadcasts. For undefined concepts we refer the reader to [5]. Dominating BroadcastsConsider a broadcast f on a connected graph G = (V, E).
For a graph G = (V, E), the k-dominating graph of G, denoted by D k (G), has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of D k (G) are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote by d 0 (G) the smallest integer for which D k (G) is connected for all k ≥ d 0 (G). It is known that Γ(G) + 1 ≤ d 0 (G) ≤ |V |, where Γ(G) is the upper domination number of G, but constructing a graph G such that d 0 (G) > Γ(G) + 1 appears to be difficult.We present two related constructions. The first construction shows that for each integer k ≥ 3 and each integer r such that 1 ≤ r ≤ k − 1, there exists a graph G k,r such that Γ(G k,r ) = k, γ(G k,r ) = r + 1 and d 0 (G k,r ) = k + r = Γ(G) + γ(G) − 1. The second construction shows that for each integer k ≥ 3 and each integer r such that 1 ≤ r ≤ k − 1, there exists a graph Q k,r such that Γ(Q k,r ) = k, γ(Q k,r ) = r and d 0 (Q k,r ) = k + r = Γ(G) + γ(G).
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