%0 Journal Article
%T Coalition Graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Haynes, Teresa W.
%A Hedetniemi, Jason T.
%A Hedetniemi, Stephen T
%A McRae, Alice
%A Mohan, Raghuveer
%D 2023
%\ 06/01/2023
%V 8
%N 2
%P 423-430
%! Coalition Graphs
%K dominating set
%K Coalition
%K independent dominating set
%R 10.22049/cco.2022.27916.1394
%X 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 $V_j$. 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. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph.
%U http://comb-opt.azaruniv.ac.ir/article_14431_a7f093b82d7cfb1bf521fdf78f7dd886.pdf