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Trees and unicyclic graphs are $$\gamma$$-graphs. (English) Zbl 1200.05162
Summary: 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 $$\gamma\cdot G$$ and called $$\gamma$$-graph of $$G$$, is defined as follows: $$V(\gamma\cdot G)$$ is the set of all $$\gamma$$-sets of $$G$$ and two sets $$D$$ and $$S$$ of $$V(\gamma\cdot G)$$ are adjacent in $$\gamma\cdot G$$ if and only if $$|D\cap S|= \gamma(G)-1$$. A graph $$G$$ is said to be $$\gamma$$-connected if $$\gamma\cdot G$$ 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.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C05 Trees
##### Keywords:
domination; $$\gamma$$-graph