id: 06062432
dt: j
an: 06062432
au: Chellali, Mustapha; Volkmann, Lutz
ti: Global offensive $k$-alliance in bipartite graphs.
so: Opusc. Math. 32, No. 1, 83-89 (2012).
py: 2012
pu: Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie, Kraków;
    Wydawnictwa AHG, Kraków
la: EN
cc: 
ut: global offensive $k$-alliance number; bipartite graphs; trees
ci: 
li: doi:10.7494/OpMath.2012.32.1.83
ab: Summary: 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 $|N (v)
    \cap S| \geq |N (v) - S| + k$ for every $v \in V(G) - S$, where $0 \leq
    k \leq Δ$ and $Δ$ is the maximum degree of $G$. The global offensive
    $k$-alliance number $γ^{k}_{o} (G)$ is the minimum cardinality of a
    global offensive $k$-alliance in $G$. We show that for every bipartite
    graph $G$ and every integer $k \geq 2, γ^{k}_{o} (G)\leq \frac
    {n(G)+|L_{k} (G)|}{2}$ , where $L_{k} (G)$ is the set of vertices of
    degree at most $k - 1$. Moreover, extremal trees attaining this upper
    bound are characterized.
rv: