Strong Alliances in Graphs

For any simple connected graph $G=(V,E)$, a defensive alliance is a subset $S$ of $V$ satisfying the condition that every vertex $vin S$ has at most one more neighbour in $V-S$ than it has in $S$. The minimum cardinality of any defensive alliance in $G$ is called the alliance number of $G$, denoted $a(G)$. In this paper, we introduce a new type of alliance number called $k$-strong alliance number and its varieties. The bounds for 1-strong alliance number in terms of different graphical parameters are determined and the characterizations of graphs with 1-strong alliance number 1, 2, and $n$ are obtained.