$\gamma_{e}$ - GRAPHS OF GRAPHS

A set $S \subseteq V$ is an equitable dominating set of a graph $G = (V,E)$ if every vertex in $V - S$ is equitably adjacent to at least one vertex in S. The equitable domination number $\gamma_{e}(G)$ of $G$ equals the minimum cardinality of an equitable dominating set S in G; we say that such a set S is a $\gamma_{e}-$set. In this paper we consider the family of all $\gamma_{e}-sets$ in a graph G and we define the $\gamma_{e}-graph ~G(\gamma_{e}) = (V (\gamma_{e}),E(\gamma_{e}))$ of $G$ to be the graph whose vertices $V (\gamma_{e})$ correspond 1-to-1 with the $\gamma_{e}-$sets of $G$, and two $\gamma_{e}-$sets, say $D_{1}$ and $D_{2}$, are adjacent in $E(\gamma_{e})$ if there exists a vertex $v \in D_{1}$ and a vertex $w \in D_{2}$ such that $v$ is adjacent to $w$ and $D_{1} = D_{2} - \{w\} \cup \{v\}$, or equivalently, $D_{2} = D_{1} - \{v\} \cup \{w\}$. In this paper we initiate the study of $\gamma_{e}-$ graph of graphs.