On connected resolvability of graphs

For an ordered set W = {w1, w2, · · · , wk} of vertices and a vertex v in a connected graph G, the 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 connected resolving set for G if distinct vertices of G have distinct representations with respect to W and the subgraph 〈W 〉 induced by W is a nontrivial connected subgraph of G. The minimum cardinality of a connected resolving set in a graph G is its connected resolving number cr(G). A connected resolving set in G of cardinality cr(G) is called a cr-set of G. An upper bound for the connected resolving number of a connected graph that is not a path is presented. We study how the connected resolving number of a connected graph is affected by adding a vertex to the graph. It is shown that for every integer k ≥ 2, there exists a connected graph with a unique cr-set. Moreover, for every pair k, r of integers with k ≥ 2 and 0 ≤ r ≤ k, there exists a connected graph G with connected resolving number k such that there are exactly r vertices in G that belong to every cr-set of G.