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 D 2 , are adjacent in E(γ) if there exists a vertex v ∈ D 1 and a vertex w ∈ D 2 such that v is adjacent to w and D 1 = D 2 − {w} ∪ {v}, or equivalently, D 2 = D 1 − {v} ∪ {w}. In this paper we initiate the study of γ-graphs of graphs.

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A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.

Let S be a set of vertices in a graph G (F, E). The authors state that a vertex u in S has a private neighbor (relative to S) if either u is not adjacent to any vertex in S or u is adjacent to a vertex w that is not adjacent to any other vertex in S. Based on the notion of private neighbors, a set of eight graph theoretic parameters can be defined whose inequality relationships can be described by a three-dimensional cube. Most of these parameters have already been studied independently. This paper unifies this study and helps to form a cohesive theory of private neighbors in graphs. Theoretical and algorithmic properties of this private neighbor cube are investigated, and many open questions are raised.

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