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

Abstract

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) $\frac{1}{2} \log (n - k+1) \leq \sinv'_k(n) \leq \log n + 4k -3$ for all $n \in \mathbb{Z}_{\geq 0}$; (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 $\frac{1}{2}\log(2k+1) \leq \sinv'_k(T) \leq \sinv_k(T)$ for some $T$; (iv) if $T$ is a tournament of order at least $28k-5$ (resp. $14k-3$), then $\sinv_k(T) \leq 1$ (resp. $\sinv_k(T) \leq 6$); (v) for every $\epsilon>0$, there exists $C$ such that $\sinv_k(T) \leq C$ for every tournament $T$ on at least $2k+1 + \epsilon k$ vertices.