SUBTOURNAMENTS ISOMORPHIC TO W₅OF AN INDECOMPOSABLE TOURNAMENT

Abstract

Abstract. We consider a tournament T = (V, A). For each subset X of V is associated the subtournament T (X) = (X, A∩(X ×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 ∈ X and x ∈ V \X, (a, x) ∈ A if and only if (b, x) ∈ A. For example, ∅, {x}(x ∈ 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 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 ∈ V for which there exists a subset X of V such that x ∈ X and T (X) is isomorphic to W 5 . We prove the following: for any indecomposable tournamentBy giving examples, we also verify that this statement is optimal.

Structure Theorem forU5-free Tournaments

“…(1), (2), and (3) follow by Claims 5 and 6. For i 0 > 1, the lower bounds of (4) and (5) follow by Claims 4 and 2.…”

“…Let G be a prime tournament which is not T n , U n , or W n for any odd n, and let H be a prime subtournament of G with 5 ≤ |V (H)| < |V (G)|. Then there exists a prime subtournament of G with |V (H)| + 1 vertices that has a subtournament isomorphic to H. This theorem can be used to prove the following result, which appears with a different proof in Belkhechine and Boudabbous [2]. Theorem 2.4 (Belkhechine and Boudabbous [2]).…”

“…Then there exists a prime subtournament of G with |V (H)| + 1 vertices that has a subtournament isomorphic to H. This theorem can be used to prove the following result, which appears with a different proof in Belkhechine and Boudabbous [2]. Theorem 2.4 (Belkhechine and Boudabbous [2]). Let G be a prime tournament which contains T 5 .…”

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

“…Notice that a critical tournament is isomorphic to its dual. Moreover, as a tournament on 4 vertices is not prime, we have: As mentioned in [2], the next remark follows from the definition of the critical tournaments.…”

“…These configurations involve mainly partially critical tournaments. We begin with the two following lemmas obtained in [2].…”