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 P3 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 P3 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 P3 convexity and we prove that it is W[1]-hard (parameterized by the size of the solution).

翻译：本文研究图的两种凸性参数：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，但其计算复杂度至今未解。一般位置数在测地凸性中定义，并于2018年被证明为NP困难。我们将这些参数推广至任意图的凸性，并证明在P3凸性中迭代数为NP困难。利用这一结果，我们进一步证明即使在直径为二的图中，测地凸性中的迭代时间亦为NP困难，这解答了一个长期悬而未决的问题。这些结果同样重要，因为它们是测地凸性与P3凸性中最常研究的十个图凸性参数中最后两个缺失的NP困难性结论。此外，我们还证明单音凸性中一般位置数在参数化复杂性上为W[1]-困难（以解的大小为参数），且对于任意ε>0，除非P=NP，否则在多项式时间内无法实现n^{1-ε}近似，即使对于直径为二的图也是如此。最后，我们得到了关于P3凸性中一般位置数的固定参数可解结果，并证明其参数化复杂性为W[1]-困难（以解的大小为参数）。