some results on domination in the generalized total graph of a commutative ring

Let r be a commutative ring with nonzero identity and h be a nonempty proper subset of r such that r/h is a saturated multiplicatively closed subset of r. anderson and badawi [4] introduced the generalized total graph of r as an undirected simple graph gth(r) with vertex set as r and any two distinct vertices x and y are adjacent if and only if x + y ∈ h. the main objective of this paper is to study the domination properties of the graph gth(r). we determine the domination number of gth(r) and its induced subgraphs gth(h) and gth(r/h). we establish a relationship between the domination number of gth(r) and the same of gth(r/h). we also establish a relationship between diameter and domination number of gth(r/h). in addition, we obtain the bondage number of gth(r). finally, a relationship between girth and bondage number of gth(r/h) has been established.