The simplicity index of tournaments

An $n$-tournament $T$ with vertex set $V$ is simple if there is no subset $M$ of $V$ such that $2\le \mid M\mid \le n-1$ and for every $x\in V\setminus M$,either$M\rightarrow x$or$x \rightarrow M$. The simplicity index of an $n$-tournament $T$ is the minimum number $s(T)$ of arcs whose reversal yields a non-simple tournament. M"{u}ller and Pelant (1974) proved that $s(T)\le \frac{n-1}{2}$, and that equality holds if and only if $T$ is doubly regular. As doubly regular tournaments exist only if $n\equiv 3\pmod{4}$,$s(T)<\frac{n-1}{2}$for$n\not\equiv3\pmod{4}$. In this paper, we study the class of $n$-tournaments with maximal simplicity index for $n\equiv ̸3(\mathrm{m}\mathrm{o}\mathrm{d}4)$.