Vertices belonging to all critical sets of a graph

Vadim E. Levit, Eugen Mandrescu

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16 Scopus citations


Let G = (V,E) be a graph. A set S V is independent if no two vertices from S are adjacent, while core(G) is the intersection of all maximum independent sets [V. E. Levit and E. Mandrescu, Discrete Appl. Math., 117 (2002), pp. 149-161]. The independence number α(G) is the cardinality of a largest independent set, and μ(G) is the size of a maximum matching of G. The neighborhood of A V is N(A) = {v V : N(v)nA = Ø}. The number dc(G) =max{|X|-|N(X)| : X V } is called the critical difference of G, and A is critical if |A| - |N(A)| = dc(G) [C. Q. Zhang, SIAM J. Discrete Math., 3 (1990), pp. 431-438]. We define ker(G) as the intersection of all critical sets. In this paper we prove that if dc(G) = 1, then ker(G) core(G) and |ker(G)| > dc(G) = a(G)-μ(G).

Original languageEnglish
Pages (from-to)399-403
Number of pages5
JournalSIAM Journal on Discrete Mathematics
Issue number1
StatePublished - 2012


  • Critical difference
  • Critical set
  • Independent set
  • Maximum matching


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