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Comptes Rendus Mathématique
Volume 351, n° 13-14
pages 501-504 (juillet 2013)
Doi : 10.1016/j.crma.2013.07.021
Received : 30 June 2013 ;  accepted : 31 July 2013
The indecomposable tournaments T with  
Les tournois indécomposables T tels que  
 

Houmem Belkhechine a , Imed Boudabbous b , Kaouthar Hzami c
a Carthage University, Bizerte Preparatory Engineering Institute, Tunisia 
b Sfax University, Sfax Preparatory Engineering Institute, Tunisia 
c Gabes University, Higher Institute of Computer Sciences and Multimedia of Gabes, Tunisia 

Abstract

We consider a tournament  . For  , the subtournament of T induced by X is  . An interval of T is a subset X of V such that, for   and  ,   if and only if  . The trivial intervals of T are ∅,   and V . A tournament is indecomposable if all its intervals are trivial. For  ,   denotes the unique indecomposable tournament defined on   such that   is the usual total order. Given an indecomposable tournament T ,   denotes the set of   such that there is   satisfying   and   is isomorphic to  . Latka [[6]] characterized the indecomposable tournaments T such that  . The authors [[1]] proved that if  , then  . In this note, we characterize the indecomposable tournaments T such that  .

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Résumé

Considérons un tournoi  . Pour  , le sous-tournoi de T induit par X est  . Un intervalle de T est une partie X de V telle que, pour tous   et  ,   si et seulement si  . Les intervalles triviaux de T sont ∅,   et V . Un tournoi est indécomposable si tous ses intervalles sont triviaux. Pour  ,   est lʼunique tournoi indécomposable défini sur   tel que   est lʼordre total usuel. Étant donné un tournoi indécomposable T ,   désigne lʼensemble des sommets   pour lesquels il existe une partie W de V telle que   et   est isomorphe à  . Latka [[6]] a caractérisé les tournois indécomposables T tels que  . Les auteurs [[1]] ont prouvé que, si  , alors  . Dans cette note, nous caractérisons les tournois indécomposables T tels que  .

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