The indecomposable tournaments $T$ with $\mid W_{5}(T) \mid = \mid T \mid -2$

TL;DR

This paper characterizes indecomposable tournaments where the set of vertices involved in a specific 5-vertex substructure is exactly two less than the total number of vertices, extending previous characterizations.

Contribution

It provides a complete characterization of indecomposable tournaments with exactly two fewer vertices involved in the W5 substructure than the total.

Findings

Characterization of indecomposable tournaments with |W5(T)| = |V| - 2

Extension of Latka's and HIK's previous results

Identification of structural properties of such tournaments

Abstract

We consider a tournament $T=(V, A)$. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X] = (X, A \cap (X \times X))$. An interval of $T$ is a subset $X$ of $V$ such that for $a, b\in X$ and $ x\in V\setminus X$, $(a,x)\in A$ if and only if $(b,x)\in A$. The trivial intervals of $T$ are $\emptyset$, $\{x\}(x\in V)$ and $V$. A tournament is indecomposable if all its intervals are trivial. For $n\geq 2$, $W_{2n+1}$ denotes the unique indecomposable tournament defined on $\{0,\dots,2n\}$ such that $W_{2n+1}[\{0,\dots,2n-1\}]$ is the usual total order. Given an indecomposable tournament $T$, $W_{5}(T)$ denotes the set of $v\in V$ such that there is $W\subseteq V$ satisfying $v\in W$ and $T[W]$ is isomorphic to $W_{5}$. Latka \cite{BJL} characterized the indecomposable tournaments $T$ such that $W_{5}(T)=\emptyset$. The authors \cite{HIK} proved that if $W_{5}(T)\neq \emptyset$, then $\mid W_{5}(T) \mid \geq \mid V \mid -2$. In this article, we characterize the indecomposable tournaments $T$ such that $\mid W_{5}(T) \mid = \mid V \mid -2$.