Alliances and Related Domination Parameters

Quorum Coloring of Maximum Cardinality in Linear Time for a Subclass of Perfect Trees

“…[1]) that admit applications to data clustering (see [9,12,15]), and the quorum classes of a graph with respect to a given quorum coloring are nothing but defensive alliances. For further reading on the well-studied topic of defensive alliances in graphs, the reader can consult the references [2,3,4,5,6,7,9,15,16,17].…”

k-STRONG DEFENSIVE ALLIANCES IN GRAPHS

“…For any v ∈ V , N (v) = {u ∈ V : uv ∈ E} is the open neighbourhood of v and N [v] = N (v) ∪ {v} is the closed neighbourhood of v. For v ∈ S, the vertices of N (v) − S are the attackers of v with respect to S and those of N [v] ∩ S are the defenders of v with respect to S. We denote |N [v] ∩ S| by Def S (v) and |N (v) − S| by Att S (v). The set S is said to be a defensive alliance [9] if Att S (v) ≤ Def S (v) ∀ v ∈ S. More results on various alliances are found in [1,2,3,4,6,7,8,10]. For k ≥ 0, a defensive alliance S is said to be a k-strong defensive alliance [4] if for any l with 1 ≤ l ≤ k, S − S l is a defensive alliance, where S l is any subset of S with l elements.…”