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 $D$ 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 $ε>0$, there exists $C$ such that $sinv’_k(T) \leq sinv_k(T) \leq C$ for every tournament $T$ on at least $2k+1 + εk$ vertices.

💡 Research Summary

The paper investigates how to strengthen the connectivity of directed graphs by means of a novel operation called inversion. Given a digraph D, inverting a vertex set X means reversing the direction of every arc whose both endpoints lie in X; applying a sequence of such inversions flips each arc exactly when it belongs to an odd number of the chosen sets. This operation cannot affect digons (pairs of opposite arcs), so only oriented graphs (digraphs without digons) can be made acyclic by inversions.

Two parameters are introduced. For a fixed integer k ≥ 1, sinv′ₖ(D) denotes the minimum number of inversions required to turn D into a k‑arc‑strong digraph (i.e., at least k arc‑disjoint directed paths cross every cut). Similarly, sinvₖ(D) is the minimum number of inversions needed to obtain a k‑strong digraph (vertex‑connectivity k). Clearly sinv′ₖ(D) ≤ sinvₖ(D) for every digraph.

Extremal bounds.
For 2k‑edge‑connected digraphs on n vertices, the authors define sinv′ₖ(n) as the maximum of sinv′ₖ(D) over all such digraphs. They prove a logarithmic sandwich:

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