On the number of vertices/edges whose deletion preserves the Kőnig–Egerváry property

Let α(G) and μ(G) denote the cardinality of a maximum independentset and the size of a maximum matching, respectively, in the graph G=(V,E). If α(G)+μ(G)=|V|, then G is aKőnig–Egerváry graph. The number d(G)=max{|A|-|N(A)|:A⊆V} is the criticaldifference of the graph G, where N(A)=v:v∈V,N(v)∩A≠∅. Every set B⊆Vsatisfying d(G)=|B|-|N(B)| is critical. Let ε(G)=|ker(G)| and ξ(G)=|core(G)|, where ker(G) is the intersection of all critical independent sets, and core(G) is the intersection of all maximum independent sets. Itis known that ker(G)⊆core(G)holds for every graph. Let us define: ϱv(G)=|{v∈V:G-v is a Kőnig–Egerváry graph }|; ϱe(G)=|{e∈E:G-e is a Kőnig–Egerváry graph }|. Clearly, ϱv(G)=|V| andϱe(G)=|E| for bipartite graphs.Unlike the bipartiteness, the property of being a Kőnig–Egerváry graphis not hereditary. In this paper, we show that (Formula presented.) for every Kőnig–Egerváry graph G.