Small k-Dominating Sets of Regular Graphs

  • William Duckworth
  • Bernard Mans
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)


A k-dominating set of a graph, G, is a set of vertices, D⊆ V(G), such that for every vertex vV(G), either vD or there exists a vertex uD that is at distance at most k from v in G. We are interested in finding k-dominating sets of small cardinality. In this paper we consider a simple, yet efficient, randomised greedy algorithm for finding a small k-dominating set of regular graphs. We analyse the average-case performance of this heuristic by analysing its performance on random regular graphs using differential equations. This, in turn, proves an upper bound on the size of a minimum k-dominating set of random regular graphs.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chang, G.J. and Nemhauser, G.L.: The k-Domination and k-Stability Problems on Graphs. TR-540, School of Operations Research and Industrial Eng., Cornell University (1982)Google Scholar
  2. 2.
    Duckworth, W.: Greedy Algorithms and Cubic Graphs. PhD thesis, Department of Mathematics and Statistics, The University of Melbourne, Australia (2001)Google Scholar
  3. 3.
    Duckworth, W. and Wormald, N.C.: Minimum Independent Dominating Sets of Random Cubic Graphs. Random Structures and Algorithms. To Appear.Google Scholar
  4. 4.
    Favaron, O., Haynes, T.W. and Slater, P.J.: Distance-k Independent Domination Sequences. Journal of Combinatorial Mathematics and Combinatorial Computing (2000) 33 225–237zbMATHMathSciNetGoogle Scholar
  5. 5.
    Haynes, T.W., Hedetniemi, S.T. and Slater, P.J.: Domination in Graphs: Advanced topics. Marcel Dekker Inc. (1998) New YorkGoogle Scholar
  6. 6.
    Janson, S., FLuczak, T. and Rucinski, A.: Random Graphs. Wiley-Interscience (2000)Google Scholar
  7. 7.
    Johnson, D.S.: Approximation Algorithms for Combinatorial problems. In: Proceedings of the 5th Annual ACM STOC, Journal of Computer and System Sciences (1994) 9 256–278Google Scholar
  8. 8.
    Kutten, S. and Peleg, D.: Fast Distributed Construction of Small k-dominating Sets and Applications. Journal of Algorithms (1998) 28(1) 40–66zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Molloy, M. and Reed, B.: The Dominating Number of a Random Cubic Graph. Random Structures and Algorithms (1995) 7(3) 209–221zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Raz, R. and Safra, S.: A Sub-Constant Error-Probability Low-Degree Test and a Sub-Constant Error-Probability PCP Characterization of NP. In: Proceedings of the 29th Annual ACM STOC (1999) 475–484 (electronic)Google Scholar
  11. 11.
    Papadimitriou, C.H. and Yannakakis, M.: Optimization, Approximation and Complexity Classes. Journal of Computer and System Sciences (1991) 43(3) 425–440zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Reed, B.: Paths, Stars and the Number Three. Combinatorics, Probability and Computing (1996) 5 277–295zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Robinson, R.W. and Wormald, N.C.: Almost All Regular Graphs are Hamiltonian. Random Structures and Algorithms (1994) 5(2) 363–374zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Wormald, N.C.: Differential Equations for Random Processes and Random Graphs. Annals of Applied Probability (1995) 5 1217–1235zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Wormald, N.C.: Models of Random Regular Graphs. In: Surveys in Combinatorics (1999) 239–298 Cambridge University PressGoogle Scholar
  16. 16.
    Wormald, N.C.: The Differential Equation Method for Random Graph Processes and Greedy Algorithms. In: Lectures on Approximation and Randomized Algorithms (1999) 73–155, PWN, WarsawGoogle Scholar
  17. 17.
    Zito, M.: Greedy Algorithms for Minimisation Problems in Random Regular Graphs. In: Proceedings of the 19th European Symposium on Algorithms. Lecture Notes in Computer Science (2001) 2161 524–536, Springer-VerlagGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • William Duckworth
    • 1
  • Bernard Mans
    • 1
  1. 1.Department of ComputingMacquarie UniversitySydneyAustralia

Personalised recommendations