Strong coalitions in graphs

For a graph $G=(V,E)$, a set D⊂ V(G) is a strong dominating set of G, if for every vertex x∈ V (G) D there is a vertex y∈ D with xy ∈ E(G) and deg(x)≤ deg(y). A strong coalition consists of two disjoint sets of vertices V₁ and V₂, neither of which is a strong dominating set but whose union V₁∪ V₂, is a strong dominating set. A vertex partition Ω=₁, V₂,..., Vₖ \ of vertices in G is a strong coalition partition, if every set Vᵢ ∈Ω either is a strong dominating set consisting of a single vertex of degree $n-1$, or is not a strong dominating set but produces a strong coalition with another set Vⱼ ∈ Ω that is not a strong dominating set. The maximum cardinality of a strong coalition partition of G is the strong coalition number of G and is denoted by $SC(G)$. In this paper, we study properties of strong coalitions in graphs.