On the minimum number of inversions to make a digraph $k$-(arc-)strong

The {\it inversion} of a set $X$ of vertices in a digraph $D$ consists of reversing the direction of all arcs of $D\langle X\rangle$. We study $sinv'_k(D)$ (resp. $sinv_k(D)$) which is the minimum number of inversions needed to transform $D$ into a $k$-arc-strong (resp. $k$-strong) digraph and $sinv'_k(n) = \max\{sinv'_k(D) \mid D~\mbox{is a $2k$-edge-connected digraph of order $n$}\}$. We show : $(i): \frac{1}{2} \log (n - k+1) \leq sinv'_k(n) \leq \log n + 4k -3$ ; $(ii):$ for any fixed positive integers $k$ and $t$, deciding whether a given oriented graph $\vec{G}$ satisfies $sinv'_k(\vec{G}) \leq t$ (resp. $sinv_k(\vec{G}) \leq t$) is NP-complete ; $(iii):$ if $T$ is a tournament of order at least $2k+1$, then $sinv'_k(T) \leq sinv_k(T) \leq 2k$, and $sinv'_k(T) \leq \frac{4}{3}k+o(k)$; $(iv):\frac{1}{2}\log(2k+1) \leq sinv'_k(T) \leq sinv_k(T)$ for some tournament $T$ of order $2k+1$; $(v):$ if $T$ is a tournament of order at least $19k-2$ (resp. $11k-2$), then $sinv'_k(T) \leq sinv_k(T) \leq 1$ (resp. $sinv_k(T) \leq 3$); $(vi):$ for every $\epsilon>0$, there exists $C$ such that $sinv'_k(T) \leq sinv_k(T) \leq C$ for every tournament $T$ on at least $2k+1 + \epsilon k$ vertices.