Coalition Graphs of Paths, Cycles, and Trees

Document Type


Publication Date



A coalition in a graph G =(V, E) consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set of G but whose union V1 ∪ V2 is a dominating set of G.A coalition partition in a graph G of order n = |V| is a vertex partition π= {V1, V2,⋯, Vk} of V such that every set Vi either is a dominating set consisting of a single vertex of degree n - 1, or is not a dominating set but forms a coalition with another set Vj which is not a dominating set. Associated with every coalition partition πof a graph G is a graph called the coalition graph of G with respect to π, denoted CG(G, π), the vertices of which correspond one-to-one with the sets V1, V2,⋯, Vk of πand two vertices are adjacent in CG(G, π) if and only if their corresponding sets in πform a coalition. In this paper we study coalition graphs, focusing on the coalition graphs of paths, cycles, and trees. We show that there are only finitely many coalition graphs of paths and finitely many coalition graphs of cycles and we identify precisely what they are. On the other hand, we show that there are infinitely many coalition graphs of trees and characterize this family of graphs.