Duality and Hereditary König-Egerváry Set-systems

A K\"onig-Egerváry graph is a graph \(G\) satisfying \(\alpha(G)+\mu(G)=|V(G)|\), where \(\alpha(G)\) is the cardinality of a maximum independent set and \(\mu(G)\) is the matching number of \(G\). Such graphs are those that admit a matching between \(V(G)-\bigcup \Gamma\) and \(\bigcap \Gamma\) where \(\Gamma\) is a set-system comprised of maximum independent sets satisfying \(|\bigcup \Gamma'|+|\bigcap \Gamma'|=2\alpha(G)\) for every set-system \(\Gamma' \subseteq \Gamma\); in order to improve this characterization of a K\"onig-Egerváry graph, we characterize \emph{hereditary K\"onig-Egerváry set-systems} (HKE set-systems, here after). An \emph{HKE} set-system is a set-system, \(F\), such that for some positive integer, \(\alpha\), the equality \(|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha\) holds for every non-empty subset, \(\Gamma\), of \(F\). We prove the following theorem: Let \(F\) be a set-system. \(F\) is an HKE set-system if and only if the equality \(|\bigcap \Gamma_1-\bigcup \Gamma_2|=|\bigcap \Gamma_2-\bigcup \Gamma_1|\) holds for every two non-empty disjoint subsets, \(\Gamma_1,\Gamma_2\) of \(F\). This theorem is applied in \cite{hke},\cite{broken}.