On the minimum number of inversions to make a digraph kk-(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

Making a tournament k-arc-strong by reversing or deorienting arcs

We prove that every tournament T=(V,A) on n2k+1 vertices can be made k-arc-strong by reversing no more than k(k+1)/2 arcs. This is best possible as the transitive tournament needs this many arcs to be reversed. We show that the number of arcs we need to reverse in order to make a tournament k-arc-strong is closely related to the number of arcs we need to reverse just to achieve in- and out-degree at least k. We also consider, for general digraphs, the operation of deorienting an arc which is not part of a 2-cycle. That is we replace an arc xy such that yx is not an arc by the 2-cycle xyx. We prove that for every tournament T on at least 2k+1 vertices, the number of arcs we need to reverse in order to obtain a k-arc-strong tournament from T is equal to the number of arcs one needs to deorient in order to obtain a k-arc-strong digraph from T. Finally, we discuss the relations of our results to related problems and conjectures