# The simplicity index of tournaments

## TL;DR

This paper investigates the maximum possible simplicity index of tournaments that are not doubly regular, especially focusing on cases where the number of vertices does not satisfy certain modular conditions.

## Contribution

It characterizes the class of tournaments with the highest simplicity index for cases where doubly regular tournaments do not exist.

## Key findings

- Maximal simplicity index achieved for specific tournament classes.
- Established bounds for simplicity index when $n
ot\equiv 3 \\pmod{4}$.
- Provided structural insights into non-doubly regular tournaments.

## Abstract

An $n$-tournament $T$ with vertex set $V$ is simple if there is no subset $M$ of $V$ such that $2\leq \left \vert M\right \vert \leq n-1$ 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 $s(T)\leq\frac{n-1}{2}$, 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 $n\not\equiv3\pmod{4}$.