A König-Egerváry graph is a graph G satisfying    where  is the cardinality of a maximum independent set and  is the matching number of G. Such graphs are those that admit a matching between  and  where G is a set-system comprised of maximum independent sets satisfying    for every set-system  in order to improve this characterization of a König-Egerváry graph, we characterize hereditary König-Egerváry set-systems (HKE set-systems, hereafter).

An HKE set-system is a set-system, F, such that for some positive integer, a, the equality  holds for every non-empty subset, G, 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  holds for every two non-empty disjoint subsets,    of F. Consequently, we get a new characterization of a König-Egerváry graph. This theorem is applied in [2] and is a key point in [1], where we get a better characterization of a König-Egerváry graph.

König-Egerváry graph, maximum independent set, hereditary König-Egerváry set-systems.