Paired coalition in graphs

A paired coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices V₁ and V₂, neither of which is a paired dominating set but whose union V₁ ∪ V₂ is a paired dominating set. A paired coalition partition (abbreviated $pc$-partition) in a graph G is a vertex partition π= V₁,V₂, ,Vₖ such that each set Vᵢ of π is not a paired dominating set but forms a paired coalition with another set Vⱼ ∈ π. The paired coalition graph PCG(G,π) of π has the vertex set v₁,v₂, ,vₖ, where vertices vᵢ and vⱼ are adjacent if and only if Vᵢ and Vⱼ form a paired coalition in G. In this paper we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs G with $PC(G)=n$ and unicyclic graphs G with $PC(G)=n-2$.