Coalition Graphs of Paths, Cycles, and Trees

A coalition in a graph G =(V, E) consists of two disjoint sets of vertices V-1 and V-2, neither of which is a dominating set of G but whose union V-1 ? V-2 is a dominating set of G.A coalition partition in a graph G of order n = |V | is a vertex partition pi = {V-1, V-2,..., V-k} of V such that every set V-i 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 V-j which is not a dominating set. Associated with every coalition partition pi of a graph G is a graph called the coalition graph of G with respect to pi, denoted CG(G, pi), the vertices of which correspond one-to-one with the sets V-1, V-2,..., V-k of pi and two vertices are adjacent in CG(G, pi) if and only if their corresponding sets in pi form 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.