Resolvability and the upper dimension of 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. A new sharp lower bound for the dimension of a graph G in terms of its maximum degree is presented. A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). A resolving set S of G is a minimal resolving set if no proper subset of S is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim+(G). The resolving number res(G) of a connected graph G is the minimum k such that every k-set W of vertices of G is also a resolving set of G. Then 1 ≤ dim(G) ≤ dim+(G) ≤ res(G) ≤ n − 1 for every nontrivial connected graph G of order n. It is shown that dim+(G) = res(G) = n − 1 if and only if G = Kn, while dim+(G) = res(G) = 2 if and only if G is a path of order at least 4 or an odd cycle. The resolving numbers and upper dimensions of some well-known graphs are determined. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dim(G) = dim+(G) = a and res(G) = b. Also, for every positive integer N, there exists a connected graph G with res(G) − dim+(G) ≥ N and dim+(G) − dim(G) ≥ N.