Skip to main content
SpringerLink
Log in

On the Independent Domination Number of Regular Graphs

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.

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

Similar content being viewed by others

References

  1. Barefoot C., Harary F., Jones K.F.: What is the difference between the domination and independent domination numbers of cubic graph?. Graphs Combin. 7(2), 205–208 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berge C.: Theory of Graphs and its Applications. Methuen & Co. Ltd., London (1962)

    MATH  Google Scholar 

  3. Cockayne E.J., FavaronO. , Li H., MacGillivray G.: The product of the independent domination numbers of a graph and its complement. Discrete Math. 90(3), 313–317 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockayne E.J., Fricke G., Mynhardt C.M.: On a Nordhaus-Gaddum type problem for independent domination. Discrete Math. 138(1-3), 199–205 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cockayne E.J., Hedetniemi S.T.: Independence graphs. Congr. Numer. X, 471–491 (1974)

    MathSciNet  Google Scholar 

  6. Cockayne E.J., Hedetniemi S.T.: Towards a theory of domination in graphs. Networks 7(3), 247–261 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cockayne E.J., Mynhardt C.M.: Independence and domination in 3-connected cubic graphs. J. Combin. Math. Combin. Comput. 10, 173–182 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Duckworth, W.,Wormald, N.: Linear programming and the worst-case analysis of greedy algorithms on cubic graphs. Electron. J. Combin. 17(1), #R177 (2010)

  9. Favaron O.: Two relations between the parameters of independence and irredundance. Discrete Math. 70(1), 17–20 (1988)

    Article  MathSciNet  Google Scholar 

  10. Gimbel J., Vestergaard P.D.: Inequalities for total matchings of graphs. Ars Combin. 39, 109–119 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Goddard W., Henning M.A.: Nordhaus-Gaddum bounds for independent domination. Discrete Math. 268, 299–302 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harutyunyan, A., Horn, P., Verstraete, J.: Independent dominating sets in graphs of girth five. Avaible at: http://www.math.ucsd.edu/jverstra/indom-final.pdf (2010)

  13. Haviland J.: On minimum maximal independent sets of a graph. Discrete Math. 94, 95–101 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Haviland J.: Independent domination in regular graphs. Discrete Math. 143, 275–280 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Haviland J.: Upper bounds for independent domination in regular graphs. Discrete Math. 307, 2643–2646 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Haynes T.W., Hedetniemi S.T., Slater P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker Inc., New York (1998)

    MATH  Google Scholar 

  17. Haynes T.W., Hedetniemi S.T., Slater P.J.: Fundamentals of Domination in Graphs. Marcel Dekker Inc., New York (1998)

    MATH  Google Scholar 

  18. Kostochka A.V.: The independent domination number of a cubic 3-connected graph can be much larger than its domination number. Graphs Combin. 9, 235–237 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lam P.C.B., Shiu W.C., Sun L.: On independent domination number of regular graphs. Discrete Math. 202, 135–144 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rabern L.: On hitting all maximum cliques with an independent set. J. Graph Theory 66, 32–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rosenfeld M.: Independent sets in regular graphs. Israel J. Math. 2, 262–272 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  22. Seifter N.: Domination and independent domination numbers of graphs. Ars Combin. 38, 119–128 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Verstraete, J.: Personal communication. (2010)

  24. Žerovnik J., Oplerova J.: A counterexample to conjecture of Barefoot, Harary, and Jones. Graphs Combin. 9, 205–207 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zverovich I.È., Zverovich V.È.: Disproof of a conjecture in the domination theory. Graphs Combin. 10, 389–396 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing and Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634-1906, USA

    Wayne Goddard

  2. Department of Mathematics, University of Johannesburg, Auckland Park, South Africa

    Michael A. Henning & Justin Southey

  3. Department of Mathematics, The University of Southern Mississippi, 118 College Drive, Hattiesburg, MS, 39406-0001, USA

    Jeremy Lyle

Authors
  1. Wayne Goddard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael A. Henning
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jeremy Lyle
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Justin Southey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael A. Henning.

Additional information

Research supported in part by the South African National Research Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goddard, W., Henning, M.A., Lyle, J. et al. On the Independent Domination Number of Regular Graphs. Ann. Comb. 16, 719–732 (2012). https://doi.org/10.1007/s00026-012-0155-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-012-0155-4

Mathematics Subject Classification

Keywords

Search

Navigation