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The indecomposable tournaments $T$ with $mid W_{5}(T) mid = mid T mid -2$ | | Houmem Belkhechine
; Imed Boudabbous
; Kaouthar Hzami
; | | Date: |
18 Jul 2013 | | Abstract: | We consider a tournament $T=(V, A)$. For $Xsubseteq V$, the subtournament of
$T$ induced by $X$ is $T[X] = (X, A cap (X imes X))$. An interval of $T$ is
a subset $X$ of $V$ such that for $a, bin X$ and $ xin Vsetminus X$,
$(a,x)in A$ if and only if $(b,x)in A$. The trivial intervals of $T$ are
$emptyset$, ${x}(xin V)$ and $V$. A tournament is indecomposable if all its
intervals are trivial. For $ngeq 2$, $W_{2n+1}$ denotes the unique
indecomposable tournament defined on ${0,dots,2n}$ such that
$W_{2n+1}[{0,dots,2n-1}]$ is the usual total order. Given an indecomposable
tournament $T$, $W_{5}(T)$ denotes the set of $vin V$ such that there is
$Wsubseteq V$ satisfying $vin W$ and $T[W]$ is isomorphic to $W_{5}$. Latka
cite{BJL} characterized the indecomposable tournaments $T$ such that
$W_{5}(T)=emptyset$. The authors cite{HIK} proved that if $W_{5}(T)
eq
emptyset$, then $mid W_{5}(T) mid geq mid V mid -2$. In this article, we
characterize the indecomposable tournaments $T$ such that $mid W_{5}(T) mid =
mid V mid -2$. | | Source: | arXiv, 1307.5027 | | Services: | Forum | Review | PDF | Favorites | |
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