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.

翻译：有向图$D$中顶点子集$X$的{\it 反转}操作是指反转$D\langle X\rangle$中所有弧的方向。我们研究$sinv'_k(D)$（相应地$sinv_k(D)$），即把$D$变换为$k$-弧强连通（相应地$k$-强连通）有向图所需的最小反转次数，并定义$sinv'_k(n) = \max\{sinv'_k(D) \mid D~\mbox{是阶为$n$的$2k$-边连通有向图}\}$。我们证明：$(i): \frac{1}{2} \log (n - k+1) \leq sinv'_k(n) \leq \log n + 4k -3$；$(ii):$ 对于任意固定的正整数$k$和$t$，判定一个满足$sinv'_k(D)<+\infty$的给定有向图$D$是否满足$sinv'_k(D) \leq t$是NP完全的；$(iii):$ 对于任意固定的正整数$k$和$t$，判定一个满足$sinv_k(D)<+\infty$的给定有向图$D$是否满足$sinv_k(D) \leq t$是NP完全的；$(iv):$ 若$T$是阶至少为$2k+1$的竞赛图，则$sinv'_k(T) \leq sinv_k(T) \leq 2k$，且$sinv'_k(T) \leq \frac{4}{3}k+o(k)$；$(v):$ 存在某个阶为$2k+1$的竞赛图$T$使得$\frac{1}{2}\log(2k+1) \leq sinv'_k(T) \leq sinv_k(T)$；$(vi):$ 若$T$是阶至少为$19k-2$（相应地$11k-2$）的竞赛图，则$sinv'_k(T) \leq sinv_k(T) \leq 1$（相应地$sinv_k(T) \leq 3$）；$(vii):$ 对于任意$\epsilon>0$，存在常数$C$，使得每个顶点数至少为$2k+1 + \epsilon k$的竞赛图$T$都满足$sinv'_k(T) \leq sinv_k(T) \leq C$。