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.

翻译：本文中，针对给定有向图$D$，我们分别研究其面向测地凸性、P3凸性和P3*凸性中的区间数与凸包数。其中，P3*凸性虽在无向版本中已有文献记载，但据我们所知，本文首次对其给出正式定义并进行系统研究。在边界估计方面，对于强连通有向图$D$且采用面向测地凸性时，我们证明$ohng(D)\leq m(D)-n(D)+2$，且存在至少一个满足$ohng(D)=m(D)-n(D)$的实例。同时，我们确定了竞赛图中三种凸性下凸包数的精确值，由此可得相应的多项式时间算法。这些结果进一步使我们能够推导出当$D$的基础图为分裂图或余二分图时，计算$ohnp(D)$的多项式时间算法。此外，我们提出一个元定理：若判定$oing(D)\leq k$或$ohng(D)\leq k$是NP难问题或参数$k$的W[i]难问题（$i\in\mathbb{Z_+^*}$），则即使$D$的基础图为二分图，上述结论依然成立。随后我们证明：即使$D$无环且基础图为二分图，判定$ohnp(D)\leq k$或$ohnps(D)\leq k$是参数$k$的W[2]难问题；即使$D$无环，判定$ohng(D)\leq k$是参数$k$的W[2]难问题；即使$D$无向圈且基础图为弦二分图，判定$oinp(D)\leq k$或$oinps(D)\leq k$是NP完全问题；即使$D$的基础图为分裂图，判定$oinp(D)\leq k$或$oinps(D)\leq k$是参数$k$的W[2]难问题。最后，我们论证了对于有界团宽度的图，借助Courcelle定理可在线性时间内计算出面向P3和P3*凸性中的区间数与凸包数。