On Rings Domination of Total Graph of Some Graph Families

Abstract

For a nontrivial connected graph G with no isolated vertex, a nonempty subset D \(\subseteq\) V (G) is a rings dominating set if D is a dominating set and for each vertex \(\upsilon\) \(\in\) V \ D is adjacent to at least two vertices in V \ D. Thus, the dominating set D of V (G) is a rings dominating set if for all \(\upsilon\) \(\in\) V \ D, \(\mid\)N(\(\upsilon\)) \(\cap\) (V \ D)\(\mid\) \(\ge\) 2. Moreover, D is called a minimum rings dominating set if D is a rings dominating set of smallest size in a given graph. The cardinality of minimum rings dominating set of G is the rings domination number of G, denoted by \(\gamma\)ri(G). Here, we determine how the minimum rings dominating set is constructed in the total graph of some graph families with the inclusion of generated conditions for this type of domination and provide their respective rings domination number.