The Directed Distance Dimension of Oriented Graphs

For a vertex $v$ of a connected oriented graph $D$ and an ordered set $W$ = \{ $w_1,$ $w_2,$ $\cdots,$ $w_k$\} of vertices of $D$, the (directed distance) representation of $v$ with respect to $W$ is the ordered $k$-tuple $r(v|W) = ( d(v, w_1), d(v, w_2), \cdots, d(v, w_k) )$, where $d(v, w_i)$ is the directed distance from $v$ to $w_i$. 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 $\dim (D) \leq n-3$ and this bound is sharp.