In this work, for a given oriented graph D, we study its interval and hull
numbers, denoted by in(D) and hn(D), respectively, in the geodetic,
P3β and P3ββ 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, we prove that hngβ(D)β€m(D)βn(D)+2 and that there is a strongly
oriented graph such that hngβ(D)=m(D)βn(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 allows us to
deduce polynomial-time algorithms to compute hnP3ββ(D) when the
underlying graph of D is split or cobipartite. Moreover, we provide a
meta-theorem by proving that if deciding whether ingβ(D)β€k or
hngβ(D)β€k is NP-hard or W[i]-hard parameterized by k, for some
iβZ+ββ, then the same holds even if the underlying graph of D
is bipartite. Next, we prove that deciding whether hnP3ββ(D)β€k or
hnP3βββ(D)β€k is W[2]-hard parameterized by k, even if the
underlying graph of D is bipartite; that deciding whether inP3ββ(D)β€k or inP3βββ(D)β€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 inP3ββ(D)β€k or inP3βββ(D)β€k is W[2]-hard
parameterized by k, even if the underlying graph of D is split. We also
argue that the interval and hull numbers in the oriented P3β and P3ββ
convexities can be computed in polynomial time for graphs of bounded tree-width
by using Courcelle's theorem