Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph Γ is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that x + uy is a unit in R. Also, \(\overline{\Gamma}\) denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as \(\overline{\Gamma}\) and characterize all γ-sets in Γ and \(\overline{\Gamma}\). Also, we obtain the bondage number of Γ. Further, we obtain the values of some domination parameters like independent, strong and weak domination numbers of \(\overline{\Gamma}\).

中文翻译：

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有限环的广义单位和单一Cayley图的控制

设R为具有非零恒等式的有限交换环，而U（R）为R所有单元的集合。该曲线图是Γ的简单无向图的顶点集- [R ，其中两个不同的顶点X和ÿ相邻当且仅当存在一个单元元件ù在Ù（[R ），使得X + UY为单位- [R 。另外，\（\ overline {\ Gamma} \）表示Γ的补码。在本文中，我们找到了Γ的支配数γ以及\（\ overline {\ Gamma} \）并表征Γ和\（\ overline {\ Gamma} \）中的所有γ集。同样，我们得到Γ的束缚数。此外，我们获得一些控制参数的值，例如\（\ overline {\ Gamma} \）的独立，强和弱控制数。