Journal of the
Korean Mathematical Society JKMS

We consider a tournament $T=(V, A)$. For each subset $X$ of $V$ is associated the subtournament $T(X) = (X, A \cap (X \times X))$ of $T$ induced by $X$. We say that a tournament $T'$ embeds into a tournament $T$ when $T'$ is isomorphic to a subtournament of $T$. Otherwise, we say that $T$ omits $T'$. A subset $X$ of $V$ is a clan of $T$ provided that for $a, b\in X$ and $ x\in V\setminus X$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are clans of $T$, called trivial clans. A tournament is indecomposable if all its clans are trivial. In 2003, B. J. Latka characterized the class $\mathcal{T}$ of indecomposable tournaments omitting a certain tournament $W_{5}$ on 5 vertices. In the case of an indecomposable tournament $T$, we will study the set $W_{5}(T)$ of vertices $x\in V$ for which there exists a subset $X$ of $V$ such that $x \in X$ and $T(X)$ is isomorphic to $W_{5}$. We prove the following: for any indecomposable tournament $T$, if $T\not\in\mathcal{T}$, then $\mid\! W_{5}(T) \!\mid \geq \mid\! V \!\mid -2$ and $\mid\! W_{5}(T) \!\mid \geq \mid\! V \!\mid -1$ if $\mid\! V \!\mid$ is even. By giving examples, we also verify that this statement is optimal.

Keywords: tournament, indecomposable, embedding, critical

MSC numbers: 05C20, 05C60