Self-coalition graphs

Abstract

A coalition in a graph $G=(V,E)$ consists of two disjoint sets $V_1$ and $V_2$ of vertices, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1 \cup V_2$ is a dominating set of $G$. A coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ 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 $V_1, V_2, \ldots, V_k$ of $\pi$ and two vertices are adjacent in $CG(G,\pi)$ if and only if their corresponding sets in $\pi$ form a coalition. The singleton partition $\pi_1$ 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 $CG(G,\pi_{1})$, where $\pi_1$ is the singleton partition of $G$. In this paper, we characterize self-coalition graphs.