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 so far 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. We use this result to prove that the iteration time is also $\NP$-hard in the geodesic convexity even in graphs with diameter two, a long standing open question. 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. We also prove that the general position number of the monophonic convexity is $W[1]$-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. Finally, we also obtain FPT results on the general position number in the $P_3$ convexity and we prove that it is $W[1]$-hard (parameterized by the size of the solution).

翻译：本文研究了图凸性的两个参数：迭代时间与一般位置数。迭代时间于1981年在测地凸性中被定义，但其计算复杂度迄今仍未解决。一般位置数同样源于测地凸性，并于2018年被证明是$\NP$-难的。我们将这些参数推广至任意图凸性，并证明在$P_3$凸性中迭代数是$\NP$-难的。利用此结果，我们进一步证明即使在直径为二的图中，测地凸性下的迭代时间也是$\NP$-难的，这是一个长期未决的开放问题。这些结果尤为重要，因为它们是测地凸性与$P_3$凸性中十个最受研究的图凸性参数里最后两个缺失的$\NP$-难度证明。我们还证明单音凸性的一般位置数是$W[1]$-难的（以解的大小为参数），且对于任意$\varepsilon>0$，除非$\P=\NP$，否则在多项式时间内无法实现$n^{1-\varepsilon}$近似，即使对直径为二的图也是如此。最后，我们获得了$P_3$凸性下一般位置数的FPT结果，并证明它是$W[1]$-难的（以解的大小为参数）。