On domination in the total torsion element graph of a module

Abstract

Let R be a commutative ring with non-zero unity and M be a unitary R-module. Let T(M) be the set of torsion elements of M. Atani and Habibi [6] introduced the total torsion element graph of M over R as an undirected graph T(Γ(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). The main objective of this paper is to study the domination properties of the graph T(Γ(M)). The domination number of T(Γ(M)) and its induced subgraphs T or(Γ(M)) and T of(Γ(M)) has been determined. Some domination parameters of T(Γ(M)) are also studied. In particular, the bondage number of T(Γ(M)) has been determined. Finally, it has been proved that T(Γ(M)) is excellent, domatically full and well covered under certain conditions.