•
•

## Turkish Journal of Mathematics

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

#### DOI

10.3906/mat-1412-72

#### Abstract

We consider a tournament $T=(V, A)$. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X] = (X, A \cap (X \times X))$. A module 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 modules of $T$ are $\emptyset$, $\{x\}(x\in V)$, and $V$. A tournament is prime if all its modules are trivial. For $n\geq 2$, $W_{2n+1}$ denotes the unique prime tournament defined on $\{0,\dots,2n\}$ such that $W_{2n+1}[\{0,\dots,2n-1\}]$ is the usual total order. Given a prime 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 $W_{5}$. B.J. Latka characterized the prime tournaments $T$ such that $W_{5}(T)=\emptyset$. The authors 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 prime tournaments $T$ such that $\mid\! W_{5}(T) \!\mid = \mid\! V \!\mid -2$.

#### Keywords

Tournament, prime, embedding, critical, partially critical

570

582

COinS