Critical Sets in Bipartite Graphs

Vadim E. Levit, Eugen Mandrescu

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Let G = (V, E) be a graph. A set I ⊆ V is independent if no two vertices from I are adjacent, and by Ind(G) we mean the family of all independent sets of G, while core(G) is the intersection of all maximum independent sets [4]. The number dc(G) = max{{pipe}I{pipe} - {pipe}N(I){pipe}: I ∈ Ind(G)} is called the critical difference of G. A set X is critical if {pipe}X{pipe} - {pipe}N(X){pipe} = dc(G) [10]. For a bipartite graph G = (A, B, E), Ore [7] defined δ0(A) = max{{pipe}X{pipe} - {pipe}N(X){pipe}: X ⊆ A}. In this paper, we prove that dc(G) = δ0(A)+δ0(B) and ker(G) = core(G) hold for every bipartite graph G = (A, B, E), where ker(G) denotes the intersection of all critical independent sets.

Original languageEnglish
Pages (from-to)543-548
Number of pages6
JournalAnnals of Combinatorics
Issue number3
StatePublished - Sep 2013
Externally publishedYes


  • critical set
  • independent set
  • matching


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