Advertisement

Critical and Maximum Independent Sets Revisited

  • Vadim E. LevitEmail author
  • Eugen Mandrescu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

Abstract

Let G be a simple graph with vertex set \(V\left( G\right) \).

A set \(S\subseteq V\left( G\right) \) is independent if no two vertices from S are adjacent, and by \(\mathrm {Ind}(G)\) we mean the family of all independent sets of G.

The number \(d\left( X\right) =\) \(\left| X\right| -\left| N(X)\right| \) is the difference of \(X\subseteq V\left( G\right) \), and a set \(A\in \mathrm {Ind}(G)\) is critical if \(d(A)=\max \{d\left( I\right) :I\in \mathrm {Ind}(G)\}\) [34].

Let us recall the following definitions:
  • \(\mathrm {core}\left( G\right) = {\displaystyle \bigcap } \left\{ S:S\textit{ is a maximum independent set}\right\} \) [16],

  • \(\mathrm {corona}\left( G\right) = {\displaystyle \bigcup } \left\{ S:S\textit{ is a maximum independent set}\right\} \) [5],

  • \(\mathrm {\ker }(G)= {\displaystyle \bigcap } \left\{ S:S\textit{ is a critical independent set}\right\} \) [18],

  • \(\mathrm {nucleus}(G)= {\displaystyle \bigcap } \left\{ S:S\textit{ is a maximum critical independent set}\right\} \) [12]

  • \(\mathrm {diadem}(G)= {\displaystyle \bigcup } \left\{ S:S\textit{ is a (maximum) critical independent set}\right\} \) [24].

In this paper we focus on interconnections between \(\ker \), core, corona, \(\mathrm {nucleus}\), and diadem.

Keywords

Independent set Critical set Ker Core Corona Diadem Matching  

Notes

Acknowledgments

The first author would like to thank the organizers of the Mathematical Optimization Theory and Operations Research Conference - MOTOR2019 for an opportunity to deliver an invited lecture on critical independent sets.

References

  1. 1.
    Ageev, A.A.: On finding critical independent and vertex sets. SIAM J. Discret. Math. 7, 293–295 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beckenbach, I., Borndörfer, R.: Hall’s and König’s theorem in graphs and hypergraphs. Discret. Math. 341, 2753–2761 (2018)CrossRefGoogle Scholar
  3. 3.
    Bhattacharya, A., Mondal, A., Murthy, T.S.: Problems on matchings and independent sets of a graph. Discret. Math. 341, 1561–1572 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonomo, F., Dourado, M.C., Durán, G., Faria, L., Grippo, L.N., Safe, M.D.: Forbidden subgraphs and the König-Egerváry property. Discret. Appl. Math. 161, 2380–2388 (2013)CrossRefGoogle Scholar
  5. 5.
    Boros, E., Golumbic, M.C., Levit, V.E.: On the number of vertices belonging to all maximum stable sets of a graph. Discret. Appl. Math. 124, 17–25 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Butenko, S., Trukhanov, S.: Using critical sets to solve the maximum independent set problem. Oper. Res. Lett. 35, 519–524 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    DeLaVina, E.: Written on the Wall II, Conjectures of Graffiti.pc. http://cms.dt.uh.edu/faculty/delavinae/research/wowII/
  8. 8.
    DeLaVina, E., Larson, C.E.: A parallel algorithm for computing the critical independence number and related sets. Ars Math. Contemp. 6, 237–245 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Deming, R.W.: Independence numbers of graphs - an extension of the König-Egerváry theorem. Discret. Math. 27, 23–33 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey, M., Johnson, D.: Computers and Intractability, 1st edn. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  11. 11.
    Jarden, A., Levit, V.E., Mandrescu, E.: Critical and maximum independent sets of a graph. Discret. Appl. Math. 247, 127–134 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jarden, A., Levit, V.E., Mandrescu, E.: Monotonic properties of collections of maximum independent sets of a graph, Order (2018). https://link.springer.com/article/10.1007/s11083-018-9461-8
  13. 13.
    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, pp. 842–850. ACM Press (2006)Google Scholar
  14. 14.
    Larson, C.E.: A note on critical independence reductions. Bull. Inst. Comb. Appl. 5, 34–46 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Larson, C.E.: The critical independence number and an independence decomposition. Eur. J. Comb. 32, 294–300 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Levit, V.E., Mandrescu, E.: Combinatorial properties of the family of maximum stable sets of a graph. Discret. Appl. Math. 117, 149–161 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Levit, V.E., Mandrescu, E.: On \(\alpha ^{+}\)-stable König-Egerváry graphs. Discret. Math. 263, 179–190 (2003)CrossRefGoogle Scholar
  18. 18.
    Levit, V.E., Mandrescu, E.: Vertices belonging to all critical independent sets of a graph. SIAM J. Discret. Math. 26, 399–403 (2012)CrossRefGoogle Scholar
  19. 19.
    Levit, V.E., Mandrescu, E.: Critical independent sets and König-Egerváry graphs. Graphs Comb. 28, 243–250 (2012)CrossRefGoogle Scholar
  20. 20.
    Levit, V.E., Mandrescu, E.: Critical sets in bipartite graphs. Ann. Comb. 17, 543–548 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Levit, V.E., Mandrescu, E.: On the structure of the minimum critical independent set of a graph. Discret. Math. 313, 605–610 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Levit, V.E., Mandrescu, E.: Critical independent sets in a graph. In: 3rd International Conference on Discrete Mathematics, 10–14 June 2013, Karnatak University, Dharwad, India (2013)Google Scholar
  23. 23.
    Levit, V.E., Mandrescu, E.: A set and collection lemma. Electron. J. Comb. 21, #P1.40 (2014)Google Scholar
  24. 24.
    Levit, V.E., Mandrescu, E.: Critical independent sets of a graph. arXiv:1407.7368 [cs.DM], 15 p. (2014)
  25. 25.
    Levit, V.E., Mandrescu, E.: Intersections and unions of critical independent sets in bipartite graphs. Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie 57, 257–260 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Levit, V.E., Mandrescu, E.: On König-Egerváry collections of maximum critical independent sets. Art Discret. Appl. Math. 2, #P1.02 (2019)CrossRefGoogle Scholar
  27. 27.
    Lorentzen, L.C.: Notes on covering of arcs by nodes in an undirected graph, Technical report ORC 66-16, Operations Research Center, University of California, Berkeley, California (1966)Google Scholar
  28. 28.
    Lovász, L., Plummer, M.D.: Matching Theory, Annals of Discrete Mathematics, vol. 29, North-Holland, Amsterdam (1986)Google Scholar
  29. 29.
    Schrijver, A.: Combinatorial Optimization. Springer, Berlin (2003)zbMATHGoogle Scholar
  30. 30.
    Short, T.M.: KE Theory & the number of vertices belonging to all maximum independent sets in a graph, M.Sc. thesis, Virginia Commonwealth University (2011)Google Scholar
  31. 31.
    Short, T.M.: On some conjectures concerning critical independent sets of a graph. Electron. J. Comb. 23, #P2.43 (2016)Google Scholar
  32. 32.
    Sterboul, F.: A characterization of the graphs in which the transversal number equals the matching number. J. Comb. Theory B 27, 228–229 (1979)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Trukhanov, S.: Novel approaches for solving large-scale optimization problems on graphs, Ph.D. thesis, University of Texas (2008)Google Scholar
  34. 34.
    Zhang, C.Q.: Finding critical independent sets and critical vertex subsets are polynomial problems. SIAM J. Discret. Math. 3, 431–438 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Ariel UniversityArielIsrael
  2. 2.Holon Institute of TechnologyHolonIsrael

Personalised recommendations