Combinatorial properties of the family of maximum stable sets of a graph

Abstract

The stability number α ( G ) of a graph G is the size of a maximum stable set of G , core (G)=⋂{S : S is a maximum stable set in G }, and ξ ( G )=| core ( G )|. In this paper we prove that for a graph G the following assertions are true: (i) if G has no isolated vertices, and ξ ( G )⩽1, then G is quasi-regularizable; (ii) if the order of G is n , and α ( G )>( n + k −min{1,| N ( core ( G ))|})/2, for some k ⩾1, then ξ ( G )⩾ k +1; moreover, if n + k −min{1,| N ( core ( G ))|} is even, then ξ ( G )⩾ k +2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ ( G )⩾1 is true whenever α ( G )> n /2. In the case of König–Egerváry graphs, i.e., for graphs enjoying the equality α ( G )+ μ ( G )= n , where μ ( G ) is the maximum size of a matching of G , we prove that | core ( G )|>| N ( core ( G ))| is a necessary and sufficient condition for α ( G )> n /2. Furthermore, for bipartite graphs without isolated vertices, ξ ( G )⩾2 is equivalent to α ( G )> n /2. We also show that Hall's Marriage Theorem is true for König–Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely for core ( G ).