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.

翻译：本文针对给定有向图$D$，研究其在地理凸性、$P_3$凸性和$P_3^*$凸性下的区间数与凸包数，分别记为${in}(D)$和${hn}(D)$。其中$P_3^*$凸性据我们所知首次在本文中正式定义并研究，尽管其无向版本在文献中已广为人知。关于界限问题，对于强连通有向图$D$，我们证明${hn_g}(D)\leq m(D)-n(D)+2$，且存在强连通有向图满足${hn_g}(D) = m(D)-n(D)$。我们还确定了竞赛图在三种凸性下凸包数的精确值，由此可推导出多项式时间算法进行计算。这些结果使我们能够推导出当$D$的基础图为分裂图或双余图时计算${hn_{P_3}}(D)$的多项式算法。此外，我们提出元定理：若判定${in_g}(D)\leq k$或${hn_g}(D)\leq k$是NP难问题或以$k$为参数的W[i]难问题（$i\in\mathbb{Z_+^*}$），则即使$D$的基础图为二分图时结论依然成立。进一步证明：即使$D$的基础图为二分图，判定${hn_{P_3}}(D)\leq k$或${hn_{P_3^*}}(D)\leq k$是参数$k$的W[2]难问题；即使$D$不含定向环且基础图为弦二分图，判定${in_{P_3}}(D)\leq k$或${in_{P_3^*}}(D)\leq k$是NP完全问题；即使$D$的基础图为分裂图，判定${in_{P_3}}(D)\leq k$或${in_{P_3^*}}(D)\leq k$是参数$k$的W[2]难问题。我们还论证了在定向$P_3$与$P_3^*$凸性下，可通过Courcelle定理在多项式时间内计算有界树宽图的区间数与凸包数。