Self-stabilizing Minimal Global Offensive Alliance Algorithm with Safe Convergence in an Arbitrary Graph

In a graph or a network G = (V, E), a set \({\mathcal{S}} \subseteq V\) is a global offensive alliance if each node \(i \in \{V-{\mathcal{S}}\}\) has \(|N[i] \cap{\mathcal{S}}| \ge |N[i]-{\mathcal{S}}|\). A global offensive alliance \({\mathcal{S}}\) is called minimal when there does not exist a node \(i \in{\mathcal{S}}\) such that the set \({\mathcal{S}}-\{i\}\) is a global offensive alliance. In this paper, we propose a new self-stabilizing algorithm for minimal global offensive alliance. It has safe convergence property under synchronous daemon in the sense that starting from an arbitrary state, it quickly converges to a global offensive alliance (a safe state) in two rounds, and then stabilizes in a minimal global offensive alliance (the legitimate state) in O(n) rounds without breaking safety during the convergence interval, where n is the number of nodes. Space requirement at each node is O(logn) bits.