Resolving Domination in Graphs

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 is its dimension dimG. A set S of vertices in G is a dominating set for G if every vertex of G that is not in S is adjacent to some vertex of S. The minimum cardinality of a dominating set is the domination number γ(G). A set of vertices of a graph G that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called the resolving domination number γr(G). In this paper, we investigate the relationship among these three parameters.