Invertibility of digraphs and tournaments

TL;DR

This work advances the theory of inversion in digraphs by (i) presenting a fixed-parameter tractable algorithm for k-Tournament-Inversion with running time O(|V|^2) for fixed k, (ii) proving NP-completeness of inv(D)≤k for general oriented graphs and related complexity results, (iii) constructing a counterexample to the dijoin conjecture and establishing a robust k-join framework that yields exact sums under certain inv conditions, (iv) relating inv(D) to cycle transversals with tight bounds inv(D)≤τ'(D) and inv(D)≤2τ(D), and (v) asymptotically determining inv(n) for n-vertex tournaments as inv(n)=(1+o(1))n, including near-matching lower and upper bounds. These findings illuminate the algorithmic and extremal structure of digraph invertibility and connect it to classical graph parameters, with implications for both theory and potential applications in network orientation problems.

Abstract

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 $(1+o(1))n$.

Theorems & Definitions (38)

• Conjecture 1: bangjensen2022inversion
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Definition 1