The inversion number of a path-reversed tournament: Resolving a conjecture of Belkhechine, Bouaziz, Boudabbous, and Pouzet

Let $D$ be a tournament and let $X\subseteq V(D)$.

The inversion of $X$ reverses all arcs whose both endpoints lie in $X$ and leaves every other arc unchanged.

A family of inversions is a decycling family if applying all of them produces an acyclic, equivalently transitive, tournament.

The inversion number $\inv(D)$ is the minimum size of such a family.

Let $Q_n$ be the tournament on $[n]$ obtained from the natural transitive tournament by reversing precisely the consecutive pairs $12,23,\ldots,(n-1)n$.

Belkhechine, Bouaziz, Boudabbous, and Pouzet conjectured in their unpublished manuscript that a natural path-reversed family has inversion number exactly $\left\lfloor(n-1)/2\right\rfloor$.

The same problem was later recorded by Bang-Jensen, da Silva, and Havet and by Alon, Powierski, Savery, Scott, and Wilmer.

In this paper we resolve this conjecture.