Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2011; 48(3): 599-626

Printed May 1, 2011

https://doi.org/10.4134/JKMS.2011.48.3.599

Copyright © The Korean Mathematical Society.

Hereditary hemimorphy of $\{-k\}$-hemimorphic tournaments for $k \geq 5$

Moncef Bouaziz, Youssef Boudabbous, and Nadia El Amri

King Saud University, King Saud University, Universit\'e de Monastir

Abstract

Let $T=(V,A)$ be a tournament. With every subset $X$ of $V$ is associated the subtournament $T[X] =(X, A \cap(X \times X))$ of $T$, induced by $X$. The dual of $T$, denoted by $T^{\ast}$, is the tournament obtained from $T$ by reversing all its arcs. Given a tournament $T^{'}=(V,A^{'})$ and a non-negative integer $k$, $T$ and $T^{'}$ are $\{-k\}$-hemimorphic provided that for all $X \subset V$, with $| X | =k$, $T[V-X]$ and $ T^{'}[V-X]$ or $T^{\ast}[V-X]$ and $ T^{'}[V-X]$ are isomorphic. The tournaments $T$ and $T^{'}$ are said to be hereditarily hemimorphic if for all subset $X$ of $ V$, the subtournaments $T[X]$ and $T^{'}[X]$ are hemimorphic. The purpose of this paper is to establish the hereditary hemimorphy of the $\{-k\}$-hemimorphic tournaments on at least $k+7$ vertices, for every $k \geq 5$.

Keywords: tournament, isomorphy, hereditary isomorphy, hemimorphy, hereditary hemimorphy

MSC numbers: 05C20, 05C60