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 .