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 G. 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 γ ᵣ(G). In this paper, we investigate the relationship among these three parameters.
No takes yet. Share an insight, caveat, or question.
Brigham et al. (2003) studied this question.