On some conjectures concerning critical independent sets of a graph

Abstract

Let G be a simple graph with vertex set V (G). A set S ⊆ V (G) is independent if no two vertices from S are adjacent. For X ⊆ V (G), the difference of X is d(X) = |X| − |N(X)| and an independent set A is critical if d(A) = max{d(X): X ⊆ V (G) is an independent setg (possibly A = ∅). Let nucleus(G) and diadem(G) be the intersection and union, respectively, of all maximum size critical independent sets in G. In this paper, we will give two new characterizations of König-Egerváry graphs involving nucleus(G) and diadem(G). We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.