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 \(2^{n-\alpha}\).