On Rings Domination of Total Graph of Some Graph Families

For a nontrivial connected graphGwith no isolated vertex, a nonempty subsetD⊆V(G) is a rings dominatingset ifDis a dominating set and for each vertexv∈VrDis adjacent to at least two vertices inVrD.Thus, the dominating setDofV(G) is a rings dominating set if for allv∈VrD,|N(v)∩(VrD)| ≥2.Moreover,Dis called a minimum rings dominating set ifDis a rings dominating set of smallest size in a givengraph. The cardinality of minimum rings dominating set ofGis the rings domination number ofG, denotedbyγri(G). Here, we determine how the minimum rings dominating set is constructed in the total graph ofsome graph families with the inclusion of generated conditions for this type of domination and provide theirrespective rings domination number.