The iteration time and the general position number in graph convexities

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).