Introduction to double coalitions in graphs

Let $G(V, E)$ be a finite, simple, isolate-free graph. A set D of vertices of a graph G with the vertex set V is a double dominating set of G, if every vertex v∈ D has at least one neighbor in D and every vertex v ∈ V D has at least two neighbors in D. A double coalition consists of two disjoint sets of vertices V₁ and V₂, neither of which is a double dominating set but their union V₁∪ V₂ is a double dominating set. A double coalition partition of a graph G is a partition Π = ₁, V₂,..., Vₖ \ of V such that no subset of Π is a double dominating set of G, but for every set Vᵢ ∈ Π, there exists a set Vⱼ ∈ Π such that Vᵢ and Vⱼ form a double coalition. In this paper, we study properties of double coalitions in graphs.