The Geodetic Number of an Oriented Graph

For two vertices $u$ and $v$ of an oriented graph $D$, the set $I(u, v)$ consists of all vertices lying on a $u-v$ geodesic or $v-u$ geodesic in $D$. If $S$ is a set of vertices of $D$, then $I(S)$ is the union of all sets $I(u,v)$ for vertices $u$ and $v$ in $S$. The geodetic number $g(D)$ is the minimum cardinality among the subsets $S$ of $V(D)$ with $I(S)=V(D)$. Several results concerning the geodetic numbers of connected oriented graphs are presented. For a nontrivial connected graph $G$, the lower orientable geodetic number $g^-(G)$ of $G$ is the minimum geodetic number among the orientations of $G$ and the upper orientable geodetic number $g^+(G)$ is the maximum such geodetic number. It is shown that $g^- (G) \neq g^+(G)$ for every connected graph of order at least 3. The lower and upper orientable geodetic numbers of several well known classes of graphs are determined. It is shown that for every two integers $n$ and $m$ with $1 \leq n-1 \leq m \leq {n \choose 2}$, there exists a connected graph $G$ of order $n$ and size $m$ such that $g^+(G) = n$.