On α+-stable König-Egerváry graphs

Vadim E. Levit, Eugen Mandrescu

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The stability number of a graph G, denoted by α(G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called α +-stable. G is a König-Egerváry graph if its order equals α(G)+μ(G), where μ(G) is the size of a maximum matching in G. In this paper, we characterize α +-stable König-Egerváry graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egerváry graph G=(V,E) of order at least two is α +-stable if and only if G has a perfect matching and |∩{V-S: S∈Ω(G)}|1 (where Ω(G) denotes the family of all maximum stable sets of G). We also show that the equality |∩{V-S: S∈Ω(G)}|=|∩{S: S∈Ω(G)}| is a necessary and sufficient condition for a König-Egerváry graph G to have a perfect matching. Finally, we describe the two following types of α +-stable König-Egerváry graphs: those with |∩{S: S∈Ω(G)}|=0 and |∩{S: S∈Ω(G)}|=1, respectively.

Original languageEnglish
Article number1-3
Pages (from-to)179-190
Number of pages12
JournalDiscrete Mathematics
Issue number1-3
StatePublished - 28 Feb 2003
Externally publishedYes


  • Blossom
  • König-Egerváry graph
  • Maximum matching
  • Maximum stable set
  • Perfect matching
  • α -stable graph


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