Skip to main content
SpringerLink
Log in

Critical sets, crowns and local maximum independent sets

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

A set \(S\subseteq V(G)\) is independent if no two vertices from S are adjacent, and by \(\mathrm {Ind}(G)\) we mean the set of all independent sets of G. A set \(A\in \mathrm {Ind}(G)\) is critical (and we write \(A\in CritIndep(G)\)) if \(\left| A\right| -\left| N(A)\right| =\max \{\left| I\right| -\left| N(I)\right| :I\in \mathrm {Ind}(G)\}\) [37], where N(I) denotes the neighborhood of I. If \(S\in \mathrm {Ind}(G)\) and there is a matching from N(S) into S, then S is a crown [1], and we write \(S\in Crown(G)\). Let \(\Psi (G)\) be the family of all local maximum independent sets of graph G, i.e., \(S\in \Psi (G)\) if S is a maximum independent set in the subgraph induced by \(S\cup N(S)\) [22]. In this paper, we present some classes of graphs where the families CritIndep(G), Crown(G), and \(\Psi (G)\) coincide and form greedoids or even more general set systems that we call augmentoids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Abu-Khzam, F.N., Fellows, M.R., Langston, M.A., Suters, W.H.: Crown structures for vertex cover kernelization. Theory Comput. Syst. 41, 411–430 (2007)

    Article  MathSciNet  Google Scholar 

  2. Ageev, A.A.: On finding critical independent and vertex sets. SIAM J. Discret. Math. 7, 293–295 (1994)

    Article  MathSciNet  Google Scholar 

  3. Bagotskaya, N.V., Levit, V.E., Losev, I.S.: A combinatorial structure ensuring applicability of the dynamic programming method. Automat. Remote Control 10, 131–139 (1990)

    MATH  Google Scholar 

  4. A. Björner, G. M. Ziegler, Introduction to greedoids, in N. White (ed.), Matroid Applications, 284–357, Cambridge University Press, 1992

  5. Boyd, E.A., Faigle, U.: An algorithmic characterization of antimatroids. Discret. Appl. Math. 28, 197–205 (1990)

    Article  MathSciNet  Google Scholar 

  6. Butenko, S., Trukhanov, S.: Using critical sets to solve the maximum independent set problem. Oper. Res. Lett. 35, 519–524 (2007)

    Article  MathSciNet  Google Scholar 

  7. Chlebík, M., Chlebíková, J.: Crown reductions for the minimum weighted vertex cover problem. Discret. Appl. Math. 156, 292–312 (2008)

    Article  MathSciNet  Google Scholar 

  8. E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc, http://cms.dt.uh.edu/faculty/delavinae/research/wowII/

  9. DeLaVina, E., Larson, C.E.: A parallel algorithm for computing the critical independence number and related sets. Ars Mathematica Contemporanea 6, 237–245 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  11. Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer-Verlag, New York (1999)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Frucht, R., Harary, F.: On the corona of two graphs. Aequationes Math. 4, 322–324 (1970)

    Article  MathSciNet  Google Scholar 

  14. Garey, M., Johnson, D.: Computers and intractability. W. H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  15. Goecke, O.: A greedy algorithm for hereditary set systems and a generalization of the Rado-Edmonds characterization of matroids. Discret. Appl. Math. 20, 39–49 (1988)

    Article  MathSciNet  Google Scholar 

  16. Golumbic, M.C., Hirst, T., Lewenstein, M.: Uniquely restricted matchings. Algorithmica 31, 139–154 (2001)

    Article  MathSciNet  Google Scholar 

  17. Y. Kempner, V. E. Levit, Correspondence between two antimatroid algorithmic characterizations, The Electronic Journal of Combinatorics 10 (2003) #R44

  18. Korte, B., Lovász, L., Schrader, R.: Greedoids. Springer-Verlag, Berlin (1991)

    Book  Google Scholar 

  19. Larson, C.E.: A note on critical independence reductions. Bull. Inst. Combin. Appl. 5, 34–46 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Levit, V.E., Mandrescu, E.: Combinatorial properties of the family of maximum stable sets of a graph. Discret. Appl. Math. 117, 149–161 (2002)

    Article  MathSciNet  Google Scholar 

  22. Levit, V.E., Mandrescu, E.: A new greedoid: the family of local maximum stable sets of a forest. Discret. Appl. Math. 124, 91–101 (2002)

    Article  MathSciNet  Google Scholar 

  23. Levit, V.E., Mandrescu, E.: Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings. Discret. Appl. Math. 132, 163–174 (2004)

    Article  MathSciNet  Google Scholar 

  24. Levit, V.E., Mandrescu, E.: Some structural properties of very well-covered graphs. Congr. Numer. 186, 97–106 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Levit, V.E., Mandrescu, E.: Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids. Discret. Appl. Math. 155, 2414–2425 (2007)

    Article  MathSciNet  Google Scholar 

  26. Levit, V.E., Mandrescu, E.: Local maximum stable set greedoids stemming from very well-covered graphs. Discret. Appl. Math. 160, 1864–1871 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  28. Levit, V.E., Mandrescu, E.: On maximum matchings in König-Egerváry graphs. Discret. Appl. Math. 161, 1635–1638 (2013)

    Article  Google Scholar 

  29. Levit, V.E., Mandrescu, E.: Crowns in bipartite graphs. Electron. Notes Discrete Math. 54, 45–50 (2016)

    Article  Google Scholar 

  30. Lorentzen, L.C.: Notes on covering of arcs by nodes in an undirected graph, Technical report ORC 66–16. University of California, Berkeley, Operations Research Center, California (1966)

    Google Scholar 

  31. Nemhauser, G.L., Trotter, L.E., Jr.: Vertex packings: structural properties and algorithms. Math. Program. 8, 232–248 (1975)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  33. Schrijver, A.: Combinatorial Optimization. Springer, Berlin (2003)

    MATH  Google Scholar 

  34. V. V. Serganova, N. V. Bagotskaya, V. E. Levit, I. S. Losev, Greedoids and the greedy algorithm, in the book: Information Transmission & Processing Systems, vol. 2, IPIT, USSR Academy of Science, Moscow, 49–52 (1988) (in Russian)

  35. Sloper, C.: Techniques in Parameterized Algorithm Design. Department of Computer Science, University of Bergen, Norway (2005). (Ph.D. thesis)

  36. 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  Google Scholar 

  37. Zhang, C.Q.: Finding critical independent sets and critical vertex subsets are polynomial problems. SIAM J. Discret. Math. 3, 431–438 (1990)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous reviewers for helpful suggestions that led to a better exposition of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ariel University, Ariel, Israel

    Vadim E. Levit

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

    Eugen Mandrescu

Authors
  1. Vadim E. Levit
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eugen Mandrescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vadim E. Levit.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Levit, V.E., Mandrescu, E. Critical sets, crowns and local maximum independent sets. J Glob Optim 83, 481–495 (2022). https://doi.org/10.1007/s10898-021-01094-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-021-01094-z

Keywords

Search

Navigation