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 $\text{inv}(D)$ 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 $12,23,\dots ,(n-1)n$. Belkhechine, Bouaziz, Boudabbous, and Pouzet conjectured in their unpublished manuscript that a natural path-reversed family has inversion number exactly $\lfloor (n-1)\mathrm{/}2\rfloor$. 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.