On the dimension of oriented graphs

For a vertex v of an oriented graph D and an ordered set W = {w(1), w(2),...,w(k)} of vertices of D, the (directed distance) representation of v with respect to W is the ordered k-tuple r(v \W) = (d(v, w(1)), d(v, w(2)),...,d(v, w(k))), where d(v, w(i)) is the directed distance from v to wi. The set W is a resolving set for D if every two distinct vertices of D have distinct representations. The minimum cardinality of a resolving set for D is the (directed distance) dimension dim(D). The dimension ratio dr(D) of an oriented graph D of order n is dim(D)/n. It is shown that for every rational number r is an element of (0, 1), there exists an oriented graph D with dr(D) = r. Furthermore, for every rational number r is an element of (0, 1/2), there exists a tournament T with dr(T) = r. The upper orientable dimension ORD(G) of an oriented graph G is the maximum value of dim(D) among the orientations D of G for which dim(D) is defined. It is shown for every positive integer k, there exists an infinite class S of graphs G with ORD(G) = k for each G is an element of S. Also, it is shown that lim(n --> infinity) ORD(K-n) = infinity.