On the hull and interval numbers of oriented graphs

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, ${P_3}$ and ${P_3^*}$ 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 ${hn_g}(D)\leq m(D)-n(D)+2$ and that there is a strongly oriented graph such that ${hn_g}(D) = m(D)-n(D)$. 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 allows us to deduce polynomial-time algorithms to compute ${hn_{P_3}}(D)$ when the underlying graph of $D$ is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ${in_g}(D)\leq k$ or ${hn_g}(D)\leq k$ 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 ${hn_{P_3}}(D)\leq k$ or ${hn_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is bipartite; that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ 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 ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ 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 $P_3$ and $P_3^*$ convexities can be computed in polynomial time for graphs of bounded tree-width by using Courcelle's theorem.