coalition graphs

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.