## 1 Introduction  
Let \(G\) be a graph and let \(v\) be a vertex of \(G\). The _open neighbourhood_ of a vertex \(v\), denoted \(N(v)\), is the set of vertices adjacent to \(v\), and the _closed neighbourhood_ of a vertex \(v\), denoted \(N[v]\), is \(N(v)\cup\{v\}\). For a subset of vertices \(S\), we say \(N(S)=\cup_{x\in S}N(x)\) and \(N[S]=S\cup N(S)\). A subset of vertices \(D\) is a _dominating set_ of \(G\) if \(N[D]=V(G)\), that is, every vertex not in \(D\) is adjacent to a vertex in \(D\). The _domination number_ of a graph \(G\), denoted \(\gamma(G)\), is the minimum cardinality of a dominating set of \(G\). A dominating set of minimum cardinality is said to be a \(\gamma\)_-set_. For a dominating set \(D\) of \(G\) and a vertex \(x\in D\), the set of \(D\)_-private neighbours_ of \(x\), denoted \(pn(x,D)\), is \(N[x]-N[D-x]\), that is, the set of vertices in the closed neighbourhood of \(x\) and not in the closed neighbourhood of any other vertex in \(D\). If \(x\in pn(x,D)\), then \(x\) is a \(D\)_-self private neighbour_ in \(D\) and if \(y\
eq x\) and \(y\in pn(x,D)\), then \(y\) is an \(D\)_-external private neighbour_ of \(x\) in \(D\). If \(D\) is a \(\gamma\)-set, then every vertex in \(D\) has a private neighbour. The \(\gamma\)_-graph_ of a graph \(G\), introduced by [PERSON] et al. [4], is a graph denoted \(G(\gamma)\), has its vertex set as the \(\gamma\)-sets of \(G\), and two \(\gamma\)-sets \(D_{1}\) and \(D_{2}\) are adjacent in \(G(\gamma)\) if there is a vertex \(u\in D_{1}\) and a vertex \(v\in D_{2}\) such that \(D_{2}=(D_{1}-\{u\})\cup\{v\}\) and \(uv\in E(G)\). Starting with \(D_{1}\), we think of making a _swap_, that is, changing vertex \(u\) for \(v\), to form \(D_{2}\). For convenience throughout the paper, we equate each node in \(G(\gamma)\) with its corresponding \(\gamma\)-set of \(G\). A slightly different model, in which \(uv\) need not be an edge in \(G\), was introduced independently by [PERSON] and [PERSON] [8]; we do not consider this model here. [PERSON] et al. [4] studied properties of \(\gamma\)-graphs, and raised the following open questions:  
1. Is \(\Delta(T(\gamma))=O(n)\) for every tree \(T\) of order \(n\)? 2. Is \(\operatorname{diam}(T(\gamma))=O(n)\) for every tree \(T\) of order \(n\)? 3. Is \(|V(T(\gamma))|\leq 2^{\gamma(T)}\) for every tree \(T\)? 4. Which graphs are \(\gamma\)-graphs of trees? 5. Which graphs are \(\gamma\)-graphs? Can you construct a graph \(H\) that is not a \(\gamma\)-graph of any graph \(G\)? 6. For which graphs \(G\) is \(G(\gamma)\cong G\)?Under what conditions is \(G(\gamma)\) a disconnected graph? The first three questions were solved by [PERSON], [PERSON], and [PERSON] [3] and the fifth question was answered by [PERSON], [PERSON], and [PERSON] [2]; the other three questions remain open. The question of which graphs are \(\gamma\)-graphs of trees was restated in a recent survey by [PERSON] and [PERSON] [5] as a primary direction of study in this area. Some \(\gamma\)-graphs of trees were determined by [PERSON] et al. [4], as well as a couple of general properties. A _stepgrid_\(SG(k)\) is the induced subgraph of the \(k\times k\) grid graph \(P_{k}\,\square\,P_{k}\) defined as follows: \(SG(k)=(V(k),E(k))\), where  
\[V(k) =\{(i,j):1\leq i,j\leq k,\ i+j\leq k+2\},\] \[E(k) =\{((i,j),(i^{\prime},j^{\prime})):i^{\prime}=i,\ j^{\prime}=j+1 ;\ i^{\prime}=i+1,\ j^{\prime}=j\}.\]  
**Proposition 1**. : _[_4_]___  
* \(K_{1,n}(\gamma)\cong K_{1}\)_._
* _For_ \(k\geq 3\)_,_ \(K_{2,n}(\gamma)\cong K_{1,2n}\)_._
* \(P_{3k}(\gamma)\cong K_{1}\)_._
* \(P_{3k+2}(\gamma)\cong P_{k+2}\)_._
* \(P_{3k+1}(\gamma)\cong SG(k+1)\)_._  
**Theorem 1**. : _[_4_]_ _Let \(T\) be a tree, and let \(x\in V(T)\) be a vertex that does not appear in any \(\gamma\)-set of \(T\). Let \(T_{1},T_{2},\ldots,T_{k}\) be the disjoint subtrees created by deleting \(x\) from \(T\), and let \(x_{i}\in T_{i}\) be the vertex in subtree \(T_{i}\) adjacent to the vertex \(x\).