Global offensive k-alliance in bipartite graphs

Abstract. Let \(k \geq 0\) be an integer. A set \(S\) of vertices of a graph \(G=(V(G),E(G))\) is called a global offensive \(k\)-alliance if \(|N(v) \cap S| \geq |N(v) \cap S|+k\) for every \(v \in V(G)-S\), where \(0 \leq k \leq \Delta\) and \(\Delta\) is the maximum degree of \(G\). The global offensive \(k\)-alliance number \(\gamma^k_o(G)\) is the minimum cardinality of a global offensive \(k\)-alliance in \(G\). We show that for every bipartite graph \(G\) and every integer \(k \geq 2\), \(\gamma^k_o(G) \leq \frac{n(G)+|L_k(G)|}{2}\), where \(L_k(G)\) is the set of vertices of degree at most \(k-1\). Moreover, extremal trees attaining this upper bound are characterized.