On resolving domination number of special family of graphs

Let G be a simple, finite, and connected graph. A dominating set D is a set of vertices such that each vertex of G is either in D or has at least one neighbor in D. The minimum cardinality of such a set is called the domination number of G, denoted by γ(G). For an ordered set W = {w1, w2, ..., wk} of vertices and a vertex v in a connected graph G, the metric representation of v with respect to W is the k-vector r(v|W ) = (d(v, w1), d(v, w2), ..., d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for G, denoted by dim(G). A resolving domination number, denoted by γr(G), is the minimum cardinality of the resolving dominating set. In this paper, we study the existence of resolving domination number of special graph and its line graph L(G), middle graph M(G), total graph T(G), and central graph C(G) of Star graph and fan graph. We have found the minimum cardinality of those special graphs.