The First Time KE is Broken up

A relevant collection is a collection, $F$, of sets, such that each set in $F$ has the same cardinality, $α(F)$. A Konig Egervary (KE) collection is a relevant collection $F$, that satisfies $|\bigcup F|+|\bigcap F|=2α(F)$. An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In \cite{jlm} and \cite{dam}, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In \cite{hke}, Jarden characterize hke collections. Let $Γ$ be a relevant collection such that $Γ-\{S\}$ is an hke collection, for every $S \in Γ$. We study the difference between $|\bigcap Γ_1-\bigcup Γ_2|$ and $|\bigcap Γ_2-\bigcup Γ_1|$, where $\{Γ_1,Γ_2\}$ is a partition of $Γ$. We get new characterizations for an hke collection and for a KE graph.

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