Monotonic Properties of Collections of Maximum Independent Sets of a Graph

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 König-Egerváry 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 ∅≠ Γ ⊆ Ω (G) is a König-Egerváry collection if |⋃Γ|+|⋂Γ|=2α(G). We conclude with the observation that for every graph G each subcollection of a König-Egerváry collection is König-Egerváry as well.