Trees and Unicyclic Graphs are \(\gamma\)-graphs

A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is said to be a dominating set of \(G\) if each \(v \in V – D\) is adjacent to at least one vertex of \(D\). The minimum cardinality of a dominating set of \(G\) is called the domination number of \(G\) and is denoted by \(\gamma(G)\). A dominating set \(D\) with cardinality \(\gamma(G)\) is called a \(\gamma\)-set of \(G\). Given a graph \(G\), a new graph, denoted by \(\gamma.G\) and called the \(\gamma\)-graph of \(G\), is defined as follows: \(V(\gamma.G)\) is the set of all \(\gamma\)-sets of \(G\) and two sets \(D\) and \(S\) of \(V(\gamma.G)\) are adjacent in \(\gamma.G\) if and only if \(|D \cap S| = \gamma(G) – 1\). A graph \(G\) is said to be \(\gamma\)-connected if \(\gamma.G\) is connected. A graph \(G\) is said to be a \(\gamma\)-graph if there exists a graph \(H\) such that \(\gamma-H\) is isomorphic to \(G\). In this paper, we show that trees and unicyclic graphs are \(\gamma\)-graphs. Also, we obtain a family of graphs which are not \(\gamma\)-graphs.