Monotonic Properties of Collections of Maximum Independent Sets of a Graph

Adi Jarden, Vadim E. Levit, Eugen Mandrescu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Pages (from-to)199-207
Number of pages9
Issue number2
StatePublished - 15 Jul 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Springer Nature B.V.


  • Core
  • Corona
  • Critical set
  • Diadem
  • Ker
  • König-Egerváry graph
  • Maximum independent set
  • Maximum matching


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