Critical independent sets of König–Egerváry graphs

Abstract

Let G be a simple graph with vertex set V ( G ) and edge set E ( G ) . For x ∈ V ( G ) , let N G ( x ) = { y ∈ V ( G ) | x y ∈ E ( G ) } and N G ( X ) = ⋃ x ∈ X N G ( x ) . For X ⊆ V ( G ) , the number d ( X ) ≔ | X | − | N G ( X ) | is called difference of X and critical difference of G denoted by d c ( G ) is max { d ( X ) ∣ X ⊆ V ( G ) } . A set S ⊆ V ( G ) is a critical independent set if S is an independent set and d ( S ) = d c ( G ) . Zhang (1990) proved that for every graph, there exists an independent set attaining the critical difference. Let Λ 0 ( G ) denote the family of all nonempty critical independent sets of G . If α ( G ) + μ ( G ) = | V ( G ) | , then G is called a König–Egerváry graph, where α ( G ) and μ ( G ) denote independence number and matching number of G , respectively. We define k e r 0 ( G ) to be the intersection of all sets in Λ 0 ( G ) and c o r e ( G ) to be the intersection of all maximum independent sets. In this paper, we characterize König–Egerváry graphs satisfying k e r 0 ( G ) = c o r e ( G ) in terms of Gallai–Edmonds Structure Theorem. As a corollary of this result, we solve an open problem posed by Jarden et al. (2018).