The Directed Distance Dimension of Oriented Graphs

For a vertex v of a connected oriented graph D and an ordered set W = {w1, w2, . . . , wk} of vertices of D, the (directed distance) representation of v with respect to W is the ordered k-tuple r(v ∣ ∣ W ) = (d(v,w1), d(v, w2), . . . , d(v, wk)), where d(v, wi) 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) of D. The dimension of a connected oriented graph need not be defined. Those oriented graphs with dimension 1 are characterized. We discuss the problem of determining the largest dimension of an oriented graph with a fixed order. It is shown that if the outdegree of every vertex of a connected oriented graph D of order n is at least 2 and dim(D) is defined, then dim(D) n − 3 and this bound is sharp.