For a vertex v of a connected oriented graph D and an ordered set W = [wi, w2,.. •, wk} of vertices of D, Ihe (directed distcince) representation of v with respect to W is the ordered fc-tuple r(v \ W) = (d(v, w1),d(v, w2 ), •.. ,d(v, wk )), where d(v, wi ) is Ihe 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 dhn(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.
By a ternary structure we mean an ordered pair $(U_0, T_0)$, where $U_0$ is a finite nonempty set and $T_0$ is a ternary relation on $U_0$. A ternary structure $(U_0, T_0)$ is called here a directed geodetic structure if there exists a strong digraph $D$ with the properties that $V(D) = U_0$ and \[ T_0(u, v, w)\quad \text{if} \text{and} \text{only} \text{if}\quad d_D(u, v) + d_D(v, w) = d_D(u, w) \] for all $u, v, w \in U_0$, where $d_D$ denotes the (directed) distance function in $D$. It is proved in this paper that there exists no sentence ${\mathbf s}$ of the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies ${\mathbf s}$.