A set S of vertices a graph is defensive alliance if, for each element , the majority its neighbours are in . We study parameterised complexity Defensive Alliance where aim to find minimum size alliance. Our main results following: (1) has been studied extensively during last twenty years, but question whether it FPT when by feedback vertex still remained open. prove that problem W[1]-hard wide...

A nonempty set S ⊂ V is a defensive k-alliance in G = (V,E), k ∈ [−Δ,Δ] ∪ Z, if for every v ∈ S, dS(v) ≥ dS̄(v) + k. A defensive k-alliance S is called exact, if S is defensive k-alliance but is no defensive (k+1)-alliance in G. In this paper we study the mathematical properties of exact defensive k-alliances in graphs. In particular, we obtain several bounds for defensive k-alliance of a graph....

We define a boundary defensive k-alliance in a graph as a set S of vertices with the property that every vertex in S has exactly k more neighbors in S than it has outside of S. In this paper we study mathematical properties of boundary defensive k-alliances. In particular, we obtain several bounds on the cardinality of every boundary defensive k-alliance. Moreover, we consider the case in which...

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 numb...

Given a simple graph G = (V,E), a subset S of the vertices is called a global defensive alliance if S is a dominating set and for every vertex v in S at least half of the vertices in the closed neighborhood of v are in S. Similarly, a subset S is called a global offensive alliance if S is a dominating set and for every vertex v not in S at least half of the vertices in the closed neighborhood o...