A set S ⊆ V is a dominating set of a graph G = (V, E) if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. In this paper we consider the family of all γ-sets in a graph G and we define the γgraph G(γ) = (V (γ), E(γ)) of G to be the graph whose vertices V (γ) correspond 1-to-1 with the γ-sets of G, and two γ-sets, say D 1 and In this paper we initiate the

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... udy of γ-graphs of graphs. 518 2. The number of isolated vertices in G[V −S]; the smaller this number, the closer S is to being a restrained dominating set, that is a set for which γ-Graphs of Graphs 519 G[V − S] has no isolated vertices. On the other hand, the larger this number, the closer S is to being a vertex cover, that is a set S of vertices such that every edge in E contains a vertex in S. The number of edges in G[S]. 4. The number of edges in G[V − S]; again, the smaller this number, the closer S is to being a vertex cover. The number of connected components in G[S] ; the smaller this number, the closer S is to being a connected dominating set, that is, a set for which G[S] is a connected subgraph. 6. The number of connected components in G[V − S]. 7. The number of vertices in V − S that are dominated more than once by vertices in S; if every vertex in V − S is dominated at least twice then S is called a 2-dominating set. 8. The number of vertices in S having an external private neighbor in V −S; the larger this number, the closer S is to being an open irredundant dominating set, that is, a set S in which every vertex has an external private neighbor in V − S.