  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 $D$ with $sinv'_k(D)<+\infty$ satisfies $sinv'_k(D) \leq t$ is
NP-complete;
  $(iii):$ for any fixed positive integers $k$ and $t$, deciding whether a
given oriented graph $D$ with $sinv_k(D)<+\infty$ satisfies $sinv_k(D) \leq t$
is NP-complete;
  $(iv):$ if $T$ is a tournament of order at least $2k+1$, then $sinv'_k(T)
\leq sinv_k(T) \leq 2k$, and $sinv'_k(T) \leq \frac{4}{3}k+o(k)$;
  $(v):\frac{1}{2}\log(2k+1) \leq sinv'_k(T) \leq sinv_k(T)$ for some
tournament $T$ of order $2k+1$;
  $(vi):$ 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. $sinv_k(T) \leq 3$);
  $(vii):$ 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.
