Tournaments with maximal decomposability

TL;DR

This paper characterizes tournaments with the highest possible decomposability index, identifying those that require the maximum number of arc reversals to become indecomposable, based on previous results about the decomposability index.

Contribution

It provides a complete characterization of tournaments with maximal decomposability index, extending prior work on the value of this index for tournaments of a given size.

Findings

Identifies tournaments with maximal decomposability index for each size

Provides structural characterization of these maximal tournaments

Extends previous bounds on decomposability index

Abstract

Given a tournament $T$, a module of $T$ is a subset $M$ of $V(T)$ such that for $x, y\in M$ and $v\in V(T)\setminus M$, $(x,v)\in A(T)$ if and only if $(y,v)\in A(T)$. The trivial modules of $T$ are $\emptyset$, $\{u\}$ $(u\in V(T))$ and $V(T)$. The tournament $T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of $T$, denoted by $\delta(T)$, is the smallest number of arcs of $T$ that must be reversed to make $T$ indecomposable. In a previous paper, we proved that for $n \geq 5$, we have $\delta(n) = \left\lceil \frac{n+1}{4} \right\rceil$, where $\delta(n)$ is the maximum of $\delta(T)$ over the tournaments $T$ with $n$ vertices. In this paper, we characterize the tournaments $T$ with $\delta$-maximal decomposability, i.e., such that $\delta(T)=\delta(\vert T\vert)$.