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Self-coalition graphs

Haynes Teresa W., Hedetniemi Jason T., Hedetniemi Stephen T., McRae Alice A., Mohan Raghuveer

Opuscula Mathematica

2023

Vol. 43, no. 2

173--183

A coalition in a graph G = (V,E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the 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} 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 π of 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 π and two vertices are adjacent in CG(G, π) if and only if their corresponding sets in π form a coalition. The singleton partition π1 of the vertex set of G is a partition of order |V |, that is, each vertex of G is in a singleton set of the partition. A graph G is called a self-coalition graph if G is isomorphic to its coalition graph CG(G, π1), where π1 is the singleton partition of G. In this paper, we characterize self-coalition graphs.