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When is G 2 a König–Egerváry Graph?

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Abstract

The independence number of a graph G, denoted by α(G), is the cardinality of a maximum independent set, and μ(G) is the size of a maximum matching in G. If α(G) + μ(G) equals its order, then G is a König–Egerváry graph. The square of a graph G is the graph G 2 with the same vertex set as in G, and an edge of G 2 is joining two distinct vertices, whenever the distance between them in G is at most two. G is a square-stable graph if it enjoys the property α(G) = α(G 2). In this paper we show that G 2 is a König–Egerváry graph if and only if G is a square-stable König–Egerváry graph.

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References

  1. Bourjolly J.M., Hammer P.L., Simeone B.: Node weighted graphs having König-Egervary property. Math. Program. Study 22, 44–63 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourjolly J.M., Pulleyblank W.R.: König-Egerváry graphs, 2-bicritical graphs and fractional matchings. Discret. Appl. Math. 24, 63–82 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deming R.W.: Independence numbers of graphs - an extension of the König-Egerváry theorem. Discret. Math. 27, 23–33 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Favaron O.: Very well-covered graphs. Discret. Math. 42, 177–187 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Korach, E., Nguyen, T., Peis, B.: Subgraph Characterization of Red/Blue-Split Graphs and König-Egerváry Graphs. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM Press. pp. 842–850 (2006)

  6. Larson C.E.: The critical independence number and an independence decomposition. Eur. J. Combin. 32, 294–300 (2011)

    Article  MATH  Google Scholar 

  7. Levit V.E., Mandrescu E.: Well-covered and König-Egerváry graphs. Congr. Numer. 130, 209–218 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Levit V.E., Mandrescu E.: Well-covered trees. Congr. Numer. 139, 102–112 (1999)

    MathSciNet  Google Scholar 

  9. Levit V.E., Mandrescu E.: On α + -stable König-Egerváry graphs. Discrete Math. 263, 179–190 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Levit V.E., Mandrescu E.: Square-stable and well-covered graphs. Acta Univ. Apulensis 10, 297–308 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Levit, V.E., Mandrescu, E.: On König-Egerváry graphs and square-stable graphs, Acta Univ. Apulensis, Special Issue 425–435 (2009)

  12. Levit V.E., Mandrescu E.: Critical independent sets and König-Egerváry graphs. Graphs Combinat. 28, 243–250 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lovász, L., Plummer, M.D.: Matching theory. Ann Discrete Math. 29, (1986)

  14. Paschos V.T., Demange M.: A generalization of König-Egerváry graphs and heuristics for the maximum independent set problem with improved approximation ratios. Eur. J. Oper. Res. 97, 580–592 (1997)

    Article  MATH  Google Scholar 

  15. Plummer M.D.: Some covering concepts in graphs. J. Combinat. Theory 8, 91–98 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  16. Randerath B., Volkman L.: Simplicial graphs and relationships to different graph invariants. Ars Combinat. 46, 211–217 (1997)

    MATH  Google Scholar 

  17. Ravindra G.: Well-covered graphs. J. Combin. Inform. Syst. Sci. 2, 20–21 (1977)

    MathSciNet  MATH  Google Scholar 

  18. Sterboul F.: A characterization of the graphs in which the transversal number equals the matching number. J. Combinat. Theory B 27, 228–229 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Mathematics, Ariel University Center of Samaria, Ariel, Israel

    Vadim E. Levit

  2. Department of Computer Science, Holon Institute of Technology, Holon, Israel

    Eugen Mandrescu

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  1. Vadim E. Levit
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  2. Eugen Mandrescu
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Correspondence to Vadim E. Levit.

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Levit, V.E., Mandrescu, E. When is G 2 a König–Egerváry Graph?. Graphs and Combinatorics 29, 1453–1458 (2013). https://doi.org/10.1007/s00373-012-1196-5

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  • DOI: https://doi.org/10.1007/s00373-012-1196-5

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