Domination in generalized unit and unitary Cayley graphs of finite rings

Abstract

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 $$\overline{\Gamma}$$.