TY  - JOUR
ID  - Haynes2023
LB  - Haynes2023
AU  - Haynes, Teresa W.
AU  - Hedetniemi, Jason T.
AU  - Hedetniemi, Stephen T.
AU  - McRae, Alice A.
AU  - Mohan, Raghuveer
TI  - Self-coalition graphs
ST  - Self-coalition graphs
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2023
VL  - 43
IS  - 2
SP  - 173
EP  - 183
DO  - https://doi.org/10.7494/OpMath.2023.43.2.173
UR  - https://doi.org/10.7494/OpMath.2023.43.2.173
AB  - 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.
KW  - coalitions in graphs
KW  - coalition partitions
KW  - coalition graphs
KW  - domination
SN  - 1232-9274
ER  -