Opuscula Math. 43, no. 2 (2023), 173-183

https://doi.org/10.7494/OpMath.2023.43.2.173

Opuscula Mathematica

# Self-coalition graphs

Teresa W. Haynes

Jason T. Hedetniemi

Stephen T. Hedetniemi

Alice A. McRae

Raghuveer Mohan

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.

Keywords: coalitions in graphs, coalition partitions, coalition graphs, domination.

Mathematics Subject Classification: 05C69.

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- Teresa W. Haynes (corresponding author)
- East Tennessee State University, Department of Mathematics and Statistics, Johnson City, TN 37614, USA
- University of Johannesburg, Department of Mathematics, Auckland Park, South Africa

- Jason T. Hedetniemi
- Department of Mathematics, Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458, USA

- Stephen T. Hedetniemi
- Professor Emeritus, Clemson University, School of Computing, Clemson, SC 29634, USA

- Alice A. McRae
- Appalachian State University, Computer Science Department, Boone, NC 28608, USA

- Raghuveer Mohan
- Appalachian State University, Computer Science Department, Boone, NC 28608, USA

- Communicated by Andrzej Żak.

- Received: 2022-07-06.
- Revised: 2022-12-14.
- Accepted: 2022-12-16.
- Published online: 2023-03-27.