Monotonic Properties of Collections of Maximum 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. The graph G is known to be a KönigEgerváry if α (G) + μ (G) = |V (G)|, where α (G) denotes the size of a maximum independent set and μ (G) is the cardinality of a maximum matching. Let Ω(G) denote the family of all maximum independent sets, and f be the function from subcollections Γ of Ω(G) to N such that f(Γ) = | ⋃ Γ|+ | ⋂ Γ|. Our main finding claims that f is ⊳-increasing, where the preorder Γ′ ⊳ Γ means that ⋃ Γ′ ⊆ ⋃ Γ and ⋂ Γ ⊆ ⋂ Γ′. Let us say that a family ∅ 6= Γ ⊆ Ω (G) is a KönigEgerváry collection if | ⋃ Γ| + | ⋂ Γ| = 2α(G). We conclude with the observation that for every graph G each subcollection of a König-Egerváry collection is KönigEgerváry as well.