On the inversion number of oriented graphs
Jørgen Bang-Jensen, Jonas Costa Ferreira da Silva, Frédéric Havet
TL;DR
This work introduces and analyzes the inversion number inv(D) of oriented graphs, the minimum number of subgraph vertex inversions required to render D acyclic, and links it to classical cycle transversals τ, τ' and cycle packing ν. It establishes core bounds inv(D) ≤ τ'(D) and inv(D) ≤ 2τ(D), and proposes the dijoin conjecture inv(L→R) = inv(L) + inv(R), proving it in several small-inversion cases and showing its equivalence to the tournament case. The paper also investigates structural operations such as augmentations and dijoins, proving inv behaves predictably under these operations in many settings (e.g., inv(n) growth, augmentations yielding inv up to 3, and intercyclic graphs with inv ≤ 4). Finally, it delineates the computational complexity landscape, proving NP-hardness for 1-Inversion and, conditionally, for general k via the dijoin conjecture, while giving polynomial-time algorithms for 1- and 2-Tournament-Inversion and showing hardness persists under bounded inversion via subdivisions. The results advance both the structural theory of inversions in oriented graphs and the algorithmic understanding of decycling problems in tournaments and related graph families.
Abstract
Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${\rm inv}(D)$, is the minimum number of inversions needed to make $D$ acyclic. Denoting by $τ(D)$, $τ' (D)$, and $ν(D)$ the cycle transversal number, the cycle arc-transversal number and the cycle packing number of $D$ respectively, one shows that ${\rm inv}(D) \leq τ' (D)$, ${\rm inv}(D) \leq 2τ(D)$ and there exists a function $g$ such that ${\rm inv}(D)\leq g(ν(D))$. We conjecture that for any two oriented graphs $L$ and $R$, ${\rm inv}(L\rightarrow R) ={\rm inv}(L) +{\rm inv}(R)$ where $L\rightarrow R$ is the dijoin of $L$ and $R$. This would imply that the first two inequalities are tight. We prove this conjecture when ${\rm inv}(L)\leq 1$ and ${\rm inv}(R)\leq 2$ and when ${\rm inv}(L) ={\rm inv}(R)=2$ and $L$ and $R$ are strongly connected. We also show that the function $g$ of the third inequality satisfies $g(1)\leq 4$. We then consider the complexity of deciding whether ${\rm inv}(D)\leq k$ for a given oriented graph $D$. We show that it is NP-complete for $k=1$, which together with the above conjecture would imply that it is NP-complete for every $k$. This contrasts with a result of Belkhechine et al. which states that deciding whether ${\rm inv}(T)\leq k$ for a given tournament $T$ is polynomial-time solvable.