For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the digraph obtained by reversing the orientations of the edges of $D$ with both endpoints in $X$. The inversion number of $D$, $\textrm{inv}(D)$, is the minimum number of inversions which can be applied in turn to $D$ to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each $k\in\mathbb{N}$ and tournament $T$, the problem of deciding whether $\textrm{inv}(T)\leq k$ is solvable in time $O_k(|V(T)|^2)$, which is tight for all $k$. In particular, the problem is fixed-parameter tractable when parameterised by $k$. On the other hand, we build on their work to prove their conjecture that for $k\geq 1$ the problem of deciding whether a general oriented graph $D$ has $\textrm{inv}(D)\leq k$ is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called 'dijoin' digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an $n$-vertex tournament is $(1+o(1))n$.

翻译：对于有向图$D$及顶点子集$X\subseteq V(D)$，在$D$上对$X$进行反转操作是指将两端点均在$X$中的边方向反转后得到的新有向图。$D$的反转数$\textrm{inv}(D)$定义为通过一系列反转操作将$D$变为无环有向图所需的最少反转次数。针对Bang-Jensen、da Silva和Havet近期提出的问题，我们证明：对任意$k\in\mathbb{N}$和竞赛图$T$，判定$\textrm{inv}(T)\leq k$是否成立的问题可在$O_k(|V(T)|^2)$时间内求解，该时间复杂度对任意$k$均为紧界。特别地，该问题在以$k$为参数时是固定参数可解的。另一方面，我们基于他们的工作证实其猜想：当$k\geq 1$时，判定一般有向图$D$是否满足$\textrm{inv}(D)\leq k$的问题是NP完全的。我们还构造了反转数等于其圈横贯数两倍的有向图，证实了Bang-Jensen、da Silva和Havet的另一猜想，同时给出反例反驳了他们关于所谓"dijoin"有向图反转数的猜想，并证明该猜想在某些特殊情形下成立。最后，我们渐近解决了该设定下的自然极值问题，改进了Belkhechine、Bouaziz、Boudabbous和Pouzet的已有界值，证明$n$顶点竞赛图的最大反转数为$(1+o(1))n$。