A dominating broadcast on a graph G = (V E) is a function : V → {0 1 diam G} such that ( ) ≤ ( ) (the eccentricity of ) for all ∈ V and such that each vertex is within distance ( ) from a vertex with ( ) > 0. The cost of a broadcast is σ ( ) = ∈V ( ), and the broadcast number γ b (G) is the minimum cost of a dominating broadcast. A set X ⊆ V (G) is said to be irredundant if each ∈ X dominates a vertex that is not dominated by any other vertex in X ; possibly = . The irredundance number ir(G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound γ b (G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir(G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for γ b . MSC:05C69, 05C70

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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For a graph G, the γ-graph of G, G(γ), is the graph whose vertices correspond to the minimum dominating sets of G, and where two vertices of G(γ) are adjacent if and only if their corresponding dominating sets in G differ by exactly two adjacent vertices. In this paper, we present several variations of the γ-graph including those using identifying codes, locating-domination, total-domination, paired-domination, and the upper-domination number. For each, we show that for any graph H, there exist infinitely many graphs whose γ-graph variant is isomorphic to H.