In this work, for a given directed graph $D$, the convexity of geodesy ${P_3 }$ and ${P_3^*}$. This last one is considered formally defined and first studied in this paper, but the undirected version is well known in the literature. On the boundary, for a strongly directed graph $D$, prove ${hn_g}(D)\leq m(D)-n(D)+2$ and ${hn_g} (D) = m(D) -n(D)$. Also determines the exact values of the hull numbers for these three convexities in the tournament. This implies a polynomial-time algorithm for computing them. These results allow us to infer polynomial-time algorithms for computing ${hn_{P_3}}(D)$ when the underlying graph of $D$ is a split or copartite graph. Furthermore, if we decide whether ${in_g}(D)\leq k$ or ${hn_g}(D)\leq k$ is NP-hard or W,[i]For some $i\in\mathbb{Z_+^*}$, for -hard parameterized by $k$, the same is true even if the underlying graph of $D$ is bipartite. It works. Next, prove that it is W that decides between ${hn_{P_3}}(D)\leq k$ and ${hn_{P_3^*}}(D)\leq k$.[2]- Even if the underlying graph of $D$ is bipartite, it is hard-parameterized by $k$. Determine if ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is NP-complete. department graph. Determine if ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is W[2]- Hard parametrized by $k$ even if the underlying graph of $D$ is partitioned. We also claim that the directed $P_3$ and $P_3^*$ convexity spacings and Hull numbers can be computed in polynomial time for a bounded tree width graph by using Courcelle’s theorem.
