Inversion dans les tournois

TL;DR

This paper studies the inversion index of tournaments, characterizing their structure, maximum distances, and obstructions, and provides explicit descriptions and universal tournaments within these classes.

Contribution

It introduces the inversion index for tournaments, characterizes critical and $(-1)$-critical tournaments via inversions, and describes classes with bounded inversion index including obstructions and universal tournaments.

Findings

Maximum distance between tournaments is at most n-1.

Inversion index bounds for n-vertex tournaments: (n-1)/2 - log2(n) ≤ i(T) ≤ n-3.

Classes with bounded inversion index are characterized by finitely many obstructions.

Abstract

We consider the transformation reversing all arcs of a subset $X$ of the vertex set of a tournament $T$. The \emph{index} of $T$, denoted by $i(T)$, is the smallest number of subsets that must be reversed to make $T$ acyclic. It turns out that critical tournaments and $(-1)$-critical tournaments can be defined in terms of inversions (at most two for the former, at most four for the latter). We interpret $i(T)$ as the minimum distance of $T$ to the transitive tournaments on the same vertex set, and we interpret the distance between two tournaments $T$ and $T'$ as the \emph{Boolean dimension} of a graph, namely the Boolean sum of $T$ and $T'$. On $n$ vertices, the maximum distance is at most $n-1$, whereas $i(n)$, the maximum of $i(T)$ over the tournaments on $n$ vertices, satisfies $\frac {n-1}{2} - \log_{2}n \leq i(n) \leq n-3$, for $n \geq 4$. Let $ \mathcal{I}_{m}^{< \omega}$ (resp. $\mathcal{I}_{m}^{\leq \omega}$) be the class of finite (resp. at most countable) tournaments $T$ such that $i(T) \leq m$. The class $\mathcal {I}_{m}^{< \omega}$ is determined by finitely many obstructions. We give a morphological description of the members of $\mathcal {I}_{1}^{< \omega}$ and a description of the critical obstructions. We give an explicit description of an universal tournament of the class $\mathcal{I}_{m}^{\leq \omega}$.