For an ordered set W={w1,w2,…,wk} of vertices and a vertex v in a connected graph G, the code of v with respect to W is the k-vector cW(v)=(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 codes with respect to W. The minimum cardinality of a resolving set for G is its dimension dim(G). Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition and coloring in graphs, or by combining resolving property with another graph-theoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs

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For an ordered set W = {w 1 , w 2 , · · · , w k } of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the, 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.

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