Global coalition sets in graphs

Let $G=(V,E)$ be a graph. A subset S ⊆ V is called a global dominating set of G, if it serves as a dominating set in both G and its complement Ḡ. We define two disjoint subsets V₁,V₂ ⊆ V to form a global coalition if neither V₁ nor V₂ individually constitutes a global dominating set, yet their union V₁ ∪ V₂ does. A global coalition partition (abbreviated as $gc$-partition) of G is a vertex partition $π$ of $V(G)$ such that for every subset Vᵢ ∈ π, there exists another subset Vⱼ ∈ π with which Vᵢ forms a global coalition. In this paper, we initiate the study of global coalition in graphs. Specifically, we prove that every graph admits a gc-partition. Additionally, we establish an upper bound on the number of global coalitions in which each member of a gc-partition can participate. We also explore the relationships between global coalition and coalition, as well as between global coalition and perfect coalition in graphs. Finally, we explore properties of $gc$-partitions in unicyclic graphs.