Maximum Induced Matchings of Random Cubic Graphs

  • William Duckworth
  • Nicholas C. Wormald
  • Michele Zito
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1858)

Abstract

In this paper we present a heuristic for finding a large induced matching \( \mathcal{M} \) of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using differential equations and obtain a lower bound on the expected size of the induced matching returned by the algorithm. The corresponding upper bound is derived by means of a direct expectation argument. We prove that \( \mathcal{M} \) asymptotically almost surely satisfies 0:2704n < |\( \mathcal{M} \) | < 0:2821n.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bollobás, B.: Random Graphs. Academic Press, London, 1985.MATHGoogle Scholar
  2. 2.
    Cameron, K.: Induced Matchings. Discrete Applied Math., 24:97–102, 1989.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Duckworth, W., Manlove, D. and Zito, M.: On the Approximability of the Maximum Induced Matching Problem. J. of Combinatorial Optimisation (Submitted).Google Scholar
  4. 4.
    Duckworth, W. and Wormald, N.C.: Minimum Independent Dominating Sets of Random Cubic Graphs. Random Structures and Algorithms (Submitted).Google Scholar
  5. 5.
    Erdös, E.: Problems and Results in Combinatorial Analysis and Graph Theory. Discrete Math., 72:81–92, 1988.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Faudree, R.J., Gyárfas, A., Schelp, R.H. and Tuza, Z.: Induced Matchings in Bipartite Graphs. Discrete Math., 78:83–87, 1989.CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Golumbic, M.C. and Laskar, R.C.: Irredundancy in Circular Arc Graphs. Discrete Applied Math., 44:79–89, 1993.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Golumbic, M.C. and Lewenstein, M.: New Results on Induced Matchings. Discrete Applied Math., 101:157–165, 2000.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Liu, J. and Zhou, H.: Maximum Induced Matchings in Graphs. Discrete Math., 170:271–281, 1997.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Steger, A. and Yu, M.: On Induced Matchings. Discrete Math., 120:291–295, 1993.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Stockmeyer, L.J. and Vazirani, V.V.: NP-Completeness of Some Generalizations of the Maximum Matching Problem. Inf. Proc. Lett., 15(1):14–19, 1982.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Wormald, N.C.: Differential Equations for Random Processes and Random Graphs. In Lectures on Approximation and Randomized Algorithms, 73–155, PWN, Warsaw, 1999. Michal Karoński and Hans-Jürgen Prömel (Eds).Google Scholar
  13. 13.
    Zito, M.: Induced Matchings in Regular Graphs and Trees. In Proceedings of the 25th International Workshop on Graph Theoretic Concepts in Computer Science, volume 1665 of Lecture Notes in Computer Science, 89–100. Springer-Verlag, 1999.CrossRefGoogle Scholar
  14. 14.
    Zito, M.: Randomised Techniques in Combinatorial Algorithmics. PhD thesis, Department of Computer Science, University of Warwick, UK, November 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • William Duckworth
    • 1
  • Nicholas C. Wormald
    • 1
  • Michele Zito
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of MelbourneAustralia
  2. 2.Department of Computer ScienceUniversity of LiverpoolUK

Personalised recommendations