Resolvability and the upper dimension of graphs

Abstract

For an ordered set W = { w 1, w 2,…, w k } 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, w 1), d( v, w 2),…, d( v, w k )), 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 = K n , 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.