Hereditary Konig Egervary Collections

Let $G$ be a simple graph with vertex set $V(G)$. A subset $S$ of $V(G)$ is independent if no two vertices from $S$ are adjacent. The graph $G$ is known to be a Konig-Egervary (KE in short) graph if $\alpha(G) + \mu(G)= |V(G)|$, where $\alpha(G)$ denotes the size of a maximum independent set and $\mu(G)$ is the cardinality of a maximum matching. Let $\Omega(G)$ denote the family of all maximum independent sets. A collection $F$ of sets is an hke collection if $|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha$ holds for every subcollection $\Gamma$ of $F$. We characterize an hke collection and invoke new characterizations of a KE graph. We prove the existence and uniqueness of a graph $G$ such that $\Omega(G)$ is a maximal hke collection. It is a bipartite graph. As a result, we solve a problem of Jarden, Levit and Mandrescu \cite{jlm}, proving that $F$ is an hke collection if and only if it is a subset of $\Omega(G)$ for some graph $G$ and $|\bigcup F|+|\bigcap F|=2\alpha(F)$. Finally, we show that the maximal cardinality of an hke collection $F$ with $\alpha(F)=\alpha$ and $|\bigcup F|=n$ is $2^{n-\alpha}$.