🤖 AI Summary
This paper investigates the minimum-cost problem of enhancing strong connectivity in directed graphs via vertex subset inversions—specifically, determining the least number of inversions, denoted sinv′ₖ(D) and sinvₖ(D), required to render a digraph k-arc-strongly connected or k-strongly connected. Employing combinatorial graph theory, extremal analysis, and computational complexity theory, we establish the first systematic bounds on these parameters. We prove that the extremal function sinv′ₖ(n) is tightly bounded by Θ(log n), revealing a fundamental logarithmic dependence of inversion cost on graph size. We further show that deciding whether sinvₖ(D) ≤ t is NP-complete. Additionally, for tournaments, we derive a sufficient condition on the number of vertices—depending only on k—under which k-strong connectivity can be guaranteed using a constant number of inversions. These results resolve a long-standing theoretical gap concerning strong connectivity augmentation under vertex subset inversion operations.
📝 Abstract
The {it inversion} of a set $X$ of vertices in a digraph $D$ consists of reversing the direction of all arcs of $Dlangle X
angle$. 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.
