Critical and maximum independent sets of a graph

Let G be a simple graph with vertex set VG. A set A⊆VG is independent if no two vertices from A are adjacent. If αG+μG=|VG|, then G is called a König–Egerváry graph (Deming, 1979; Sterboul, 1979), where αG is the size of a maximum independent set and μG stands for the cardinality of a largest matching in G. The number dX=X−N(X) is the difference of X⊆VG, and a set A⊆VG is critical if d(A)=max{dX:X⊆VG} (Zhang, 1990). In this paper, we present various connections between unions and intersections of maximum and/or critical independent sets of a graph, which lead to new characterizations of König–Egerváry graphs.