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Key Domination Parameters

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From Domination to Coloring

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Abstract

In this chapter, two key domination parameters different from the standard domination number are discussed. The first, the independent domination number, was introduced by Stephen Hedetniemi and his coauthor Ernie Cockayne in 1974. The second, the total domination number, was introduced by Hedetniemi together with his coauthors Cockayne and Dawes in 1980.

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References

  1. H. Aram, S.M. Sheikholeslami, L. Volkmann, On the total domatic number of regular graphs. Trans. Comb. 1, 45–51 (2012)

    MathSciNet  MATH  Google Scholar 

  2. D. Archdeacon, J. Ellis-Monaghan, D. Fischer, D. Froncek, P.C.B. Lam, S. Seager, B. Wei, R. Yuster, Some remarks on domination. J. Graph Theory 46, 207–210 (2004)

    Article  MathSciNet  Google Scholar 

  3. C. Berge, Theory of Graphs and its Applications (Methuen, London, 1962)

    MATH  Google Scholar 

  4. R.C. Brigham, J.R. Carrington, R.P. Vitray, Connected graphs with maximum total domination number. J. Combin. Comput. Combin. Math. 34, 81–96 (2000)

    MathSciNet  MATH  Google Scholar 

  5. B. Chen, J.H. Kim, M. Tait, J. Verstraete, On coupon colorings of graphs. Discret. Appl. Math. 193, 94–101 (2015)

    Article  MathSciNet  Google Scholar 

  6. V. Chvátal, C. McDiarmid, Small transversals in hypergraphs. Combinatorica 12, 19–26 (1992)

    Article  MathSciNet  Google Scholar 

  7. E.J. Cockayne, S.T. Hedetniemi, Independence graphs, in Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic University, Boca Raton, FL, 1974), pp. 471–491. Congress. Numer., No. X, Utilitas Math., Winnipeg, Man. 1974

    Google Scholar 

  8. E.J. Cockayne, R.M. Dawes, S.T. Hedetniemi, Total domination in graphs. Networks 10(3), 211–219 (1977)

    Article  MathSciNet  Google Scholar 

  9. E.J. Cockayne, S.T. Hedetniemi, Disjoint independent dominating sets in graphs. Discret. Math. 15(3), 213–222 (1976)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. E.J. Cockayne, C.M. Mynhardt, The sequence of upper and lower domination, independence and irredundance numbers of a graph. Discret. Math. 122, 89–102 (1993)

    Article  MathSciNet  Google Scholar 

  12. E.J. Cockayne, S.T. Hedetniemi, D.J. Miller, Properties of hereditary hypergraphs and middle graphs. Canad. Math. Bull. 21, 461–468 (1978)

    Article  MathSciNet  Google Scholar 

  13. A. Eustis, M.A. Henning, A. Yeo, Independence in \(5\)-uniform hypergraphs. Discret. Math. 339, 1004–1027 (2016)

    Article  MathSciNet  Google Scholar 

  14. O. Favaron, Two relations between the parameters of independence and irredundance. Discret. Math. 70, 17–20 (1988)

    Article  MathSciNet  Google Scholar 

  15. W. Goddard, M.A. Henning, Independent domination in graphs: A survey and recent results. Discret. Math. 313, 839–854 (2013)

    Article  MathSciNet  Google Scholar 

  16. W. Goddard, M.A. Henning, Thoroughly dispersed colorings. J. Graph Theory 88(1), 174–191 (2018)

    Article  MathSciNet  Google Scholar 

  17. W. Goddard, M.A. Henning, Acyclic total dominating sets in cubic graphs. To appear in Appl. Anal. Discr. Math.

    Google Scholar 

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

    MATH  Google Scholar 

  19. P. Heggernes, J.A. Telle, Partitioning graphs into generalized dominating sets. Nordic J. Comput. 5, 128–142 (1998)

    MathSciNet  MATH  Google Scholar 

  20. M.A. Henning, Graphs with large total domination number. J. Graph Theory 35(1), 21–45 (2000)

    Article  MathSciNet  Google Scholar 

  21. M.A. Henning, A. Yeo, A transition from total domination in graphs to transversals in hypergraphs. Quaest. Math. 30, 417–436 (2007)

    Article  MathSciNet  Google Scholar 

  22. M.A. Henning, A. Yeo, Hypergraphs with large transversal number and with edge sizes at least three. J. Graph Theory 59, 326–348 (2008)

    MathSciNet  MATH  Google Scholar 

  23. M.A. Henning, A. Yeo, Total domination in graphs (Springer Monographs in Mathematics, 2013). ISBN: 978-1-4614-6524-9 (Print) 978-1-4614-6525-6 (Online)

    Google Scholar 

  24. O. Ore, Theory of graphs. Am. Math. Soc. Transl. 38, 206–212 (1962)

    MATH  Google Scholar 

  25. L. Sun, J. Wang, An upper bound for the independent domination number. J. Combin. Theory Ser. B 76, 240–246 (1999)

    Article  MathSciNet  Google Scholar 

  26. S. Thomassé, A. Yeo, Total domination of graphs and small transversals of hypergraphs. Combinatorica 27, 473–487 (2007)

    Article  MathSciNet  Google Scholar 

  27. Zs. Tuza, Covering all cliques of a graph. Discret. Math. 86, 117–126 (1990)

    Google Scholar 

  28. B. Zelinka, Adomatic and idomatic numbers of graphs. Math. Slovaca 33, 99–103 (1983)

    MathSciNet  MATH  Google Scholar 

  29. B. Zelinka, Total domatic number and degrees of vertices of a graph. Math. Slovaca 39, 7–11 (1989)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Gary Chartrand & Ping Zhang

  2. Department of Mathematics, East Tennessee State University, Johnson City, TN, USA

    Teresa W. Haynes

  3. Department of Mathematics, University of Johannesburg, Johannesburg, South Africa

    Michael A. Henning

Authors
  1. Gary Chartrand
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  2. Teresa W. Haynes
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  3. Michael A. Henning
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  4. Ping Zhang
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Correspondence to Gary Chartrand .

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Chartrand, G., Haynes, T.W., Henning, M.A., Zhang, P. (2019). Key Domination Parameters. In: From Domination to Coloring. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-31110-0_2

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