TY  - JOUR
ID  - Chellali2012
LB  - Chellali2012
AU  - Chellali, Mustapha
AU  - Volkmann, Lutz
TI  - Global offensive k-alliance in bipartite graphs
ST  - Global offensive k-alliance in bipartite graphs
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2012
VL  - 32
IS  - 1
SP  - 83
EP  - 89
DO  - http://dx.doi.org/10.7494/OpMath.2012.32.1.83
UR  - http://dx.doi.org/10.7494/OpMath.2012.32.1.83
AB  - 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.
KW  - global offensive \(k\)-alliance number
KW  - bipartite graphs
KW  - trees
SN  - 1232-9274
ER  -