In this paper, we study two graph convexity parameters: iteration time and general position number. The iteration time was defined in 1981 in the geodesic convexity, but its computational complexity was still open. The general position number was defined in the geodesic convexity and proved NP-hard in 2018. We extend these parameters to any graph convexity and prove that the iteration number is NP-hard in the $P_3$ convexity and, with this result, we can prove that the iteration time is also NP-hard in the geodesic convexity even in graphs with diameter two, a very natural question which was unsolved since 1981. These results are also important, since they are the last two missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and $P_3$ convexities. Finally, we also prove that the general position number of the monophonic convexity is NP-hard, W[1]-hard (parameterized by the size of the solution) and $n^{1-\varepsilon}$-inapproximable in polynomial time for any $\varepsilon>0$ unless P=NP, even in graphs with diameter two.

翻译：本文研究了图凸性中的两个参数：迭代时间和一般位置数。迭代时间于1981年在测地凸性中被定义，但其计算复杂度问题至今仍悬而未决。一般位置数同样在测地凸性中提出，并于2018年被证明为NP难问题。我们将这些参数推广至任意图凸性，并证明迭代数在$P_3$凸性中是NP难的。基于该结果，我们进而证明即使在直径为2的图中，迭代时间在测地凸性中也是NP难的——这一自1981年以来未解的自然问题至此得到解答。这些结论具有重要价值，因为它们补全了测地凸性和$P_3$凸性中最常研究的十个图凸性参数中最后两个缺失的NP难性结果。最后，我们还证明单音凸性中的一般位置数是NP难的、W[1]难的（以解的大小为参数），并且在多项式时间内对任意$\varepsilon>0$具有$n^{1-\varepsilon}$不可近似性（除非P=NP），即使对直径为2的图也是如此。