Global offensive k-alliance in bipartite graphs

• Main Authors: Mustapha Chellali, Lutz Volkmann
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2012-01-01
• Series: Opuscula Mathematica
• Subjects: global offensive \(k\)-alliance number; bipartite graphs; trees
• Online Access: http://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3207.pdf

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.