Resolvability in graphs and the metric dimension of a graph

Abstract

For an ordered subset W={ w 1, w 2,…, w k } of vertices in a connected graph G and a vertex v of 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 )). The set W is a resolving set for G if r(u | W)=r(v | W) implies that u= v for all pairs u, v of vertices of G. The metric dimension dim( G) of G is the minimum cardinality of a resolving set for G. Bounds on dim( G) are presented in terms of the order and the diameter of G. All connected graphs of order n having dimension 1, n−2, or n−1 are determined. A new proof for the dimension of a tree is also presented. From this result sharp bounds on the metric dimension of unicyclic graphs are established. It is shown that dim( H)⩽dim( H× K 2)⩽dim( H)+1 for every connected graph H. Moreover, it is shown that for every positive real number ε, there exists a connected graph G and a connected induced subgraph H of G such that dim( G)/dim( H)< ε.