In this paper, we investigate two graph convexity parameters: the iteration time and the general position number. Harary and Nieminem introduced in 1981 the iteration time in the geodesic convexity, but its computational complexity was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position number of the geodesic convexity and proved that it is NP-hard to compute. In this paper, we extend these parameters to the P3 convexity and prove that it is NP-hard to compute them. With this, we also prove that the iteration number is NP-hard on the geodesic convexity even in graphs with diameter two. These results are the last three missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities.

翻译：本文研究了两种图凸性参数：迭代时间与一般位置数。Harary与Nieminem于1981年在测地线凸性中引入了迭代时间概念，但其计算复杂度问题一直悬而未决。Manuel与Klavžar于2018年在测地线凸性中定义了一般位置数，并证明其计算是NP难的。本文将上述参数拓展至P3凸性，并证明在此凸性中计算这些参数也是NP难的。据此，我们进一步证明，即便在直径为2的图中，测地线凸性中的迭代数计算同样是NP难的。这些结果填补了测地线凸性与P3凸性中十项最受关注的图凸性参数计算复杂度的最后三项NP难性质空白。