Domination in generalized unit and unitary Cayley graphs of finite rings

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, Γ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Gamma}$$\end{document} denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as Γ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Gamma}$$\end{document} and characterize all γ-sets in Γ and Γ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Gamma}$$\end{document}. 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 Γ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Gamma}$$\end{document}.