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Let G = (V, E) be a graph. A global secure set SD ⊆ V is a dominating set which satisfies the condition: for all X ⊆ SD, |N[X] ∩ SD| ≥ | N[X] − SD|. A global defensive alliance is a set of vertices A that is dominating and satisfies a weakened condition: for all x ∈ A, |N[x] ∩ A| ≥ |N[x] − A|. We give an upper bound on the cardinality of minimum global secure sets in cactus trees. We also present some results for trees, and we relate them to the known bounds on the minimum cardinality of global defensive alliances.

Bounds on global secure sets in cactus trees