  In this work, for a given oriented graph $D$, we study its interval and hull
numbers, respectively, in the oriented geodetic, P3 and P3* convexities. This
last one, we believe to be formally defined and first studied in this paper,
although its undirected version is well-known in the literature.
  Concerning bounds, for a strongly oriented graph D, and the oriented geodetic
convexity, we prove that $ohng(D)\leq m(D)-n(D)+2$ and that there is at least
one such that $ohng(D) = m(D)-n(D)$. We also determine exact values for the
hull numbers in these three convexities for tournaments, which imply
polynomial-time algorithms to compute them. These results allow us to deduce
polynomial-time algorithms to compute $ohnp(D)$ when the underlying graph of
$D$ is split or cobipartite.
  Moreover, we provide a meta-theorem by proving that if deciding whether
$oing(D)\leq k$ or $ohng(D)\leq k$ is NP-hard or W[i]-hard parameterized by
$k$, for some $i\in\mathbb{Z_+^*}$, then the same holds even if the underlying
graph of $D$ is bipartite. Next, we prove that deciding whether $ohnp(D)\leq k$
or $ohnps(D)\leq k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic
and its underlying graph is bipartite; that deciding whether $ohng(D)\leq k$ is
W[2]-hard parameterized by $k$, even if $D$ is acyclic; that deciding whether
$oinp(D)\leq k$ or $oinps(D)\leq k$ is NP-complete, even if $D$ has no directed
cycles and the underlying graph of $D$ is a chordal bipartite graph; and that
deciding whether $oinp(D)\leq k$ or $oinps(D)\leq k$ is W[2]-hard parameterized
by $k$, even if the underlying graph of $D$ is split.
  Finally, also argue that the interval and hull numbers in the oriented P3 and
P3* convexities can be computed in cubic time for graphs of bounded
clique-width by using Courcelle's theorem.
