SUBTOURNAMENTS ISOMORPHIC TO W5 OF AN INDECOMPOSABLE TOURNAMENT

Title & Authors
SUBTOURNAMENTS ISOMORPHIC TO W5 OF AN INDECOMPOSABLE TOURNAMENT
Belkhechine, Houmem; Boudabbous, Imed; Hzami, Kaouthar;

Abstract
We consider a tournament T = (V,A). For each subset X of V is associated the subtournament T(X) = (X,$\small{A{\cap}(X{\times}X)}$) 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, $\small{b{\in}X}$ and $\small{x{\in}V{\backslash}X}$, $\small{(a,x){\in}A}$ if and only if $\small{(b,x){\in}A}$. For example, $\small{{\emptyset}}$, $\small{\{x\}(x{\in}V)}$ 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 $\small{{\tau}}$ of indecomposable tournaments omitting a certain tournament $\small{W_5}$ on 5 vertices. In the case of an indecomposable tournament T, we will study the set $\small{W_5}$(T) of vertices $\small{x{\in}V}$ for which there exists a subset X of V such that $\small{x{\in}X}$ and T(X) is isomorphic to $\small{W_5}$. We prove the following: for any indecomposable tournament T, if $\small{T{\notin}{\tau}}$, then $\small{{\mid}W_5(T){\mid}{\geq}{\mid}V{\mid}}$ -2 and $\small{{\mid}W_5(T){\mid}{\geq}{\mid}V{\mid}}$ -1 if $\small{{\mid}V{\mid}}$ is even. By giving examples, we also verify that this statement is optimal.
Keywords
tournament;indecomposable;embedding;critical;
Language
English
Cited by
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