@Article{Chellali2012,
author = "Chellali, Mustapha and Volkmann, Lutz",
title = "Global offensive k-alliance in bipartite graphs",
journal = "Opuscula Math.",
year = "2012",
volume = "32",
number = "1",
pages = "83--89",
doi = "http://dx.doi.org/10.7494/OpMath.2012.32.1.83",
url = "http://dx.doi.org/10.7494/OpMath.2012.32.1.83",
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.",
keywords = "global offensive \(k\)-alliance number, bipartite graphs, trees",
issn = "1232-9274"
}