Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.

A secure (total ) dominating set of a graph G = (V, E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V − X, there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total ) domination number γ s (G) (γ st (G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then γ st (G) ≤ γ s (G). We also show that γ st (G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

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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 permutation π of the vertex set of a graph G, the graph πG is obtained from two disjoint copies G 1 andWe study graphs for which γ(K 2 × G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V (G) and those for which γ(πG) = 2γ(G) for each permutation π of V (G).