Duality and Hereditary K\"onig-Egerv\'ary Set-systems

A K\"onig-Egerv\'ary 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-Egerv\'ary graph, we characterize \emph{hereditary K\"onig-Egerv\'ary 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}.