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.