For an ordered subset W = (w(1),w(2),...,w(lambda)) of vertices in a connec
ted graph G and a vertex v Of G, the metric representation of v with respec
t to W is the k-vector r(v \ W) = (d(v,w(1)), d(v, w(2)),...,d(v,w lambda))
, 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 pre
sented 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 th
at dim(H)less than or equal to dim(H x K-2)less than or equal to dim(H) + 1
for every connected graph H. Moreover, it is shown that for every positive
real number epsilon, there exists a connected graph G and a connected indu
ced subgraph H of G such that dim(G)/dim(H) < epsilon. (C) 2000 Elsevier Sc
ience B.V. All rights reserved.