Periodica Mathematica Hungarica

, Volume 65, Issue 1, pp 125–134 | Cite as

Nordhaus-Gaddum results for the convex domination number of a graph

  • M. Lemańska
  • J. A. Rodríguez-Velázquez
  • I. Gonzalez Yero
Article

Abstract

The distance dG(u, v) between two vertices u and v in a connected graph G is the length of the shortest uv-path in G. A uv-path of length dG(u, v) is called a uv-geodesic. A set X is convex in G if vertices from all ab-geodesics belong to X for any two vertices a, bX. The convex domination number γcon(G) of a graph G equals the minimum cardinality of a convex dominating set. In the paper, Nordhaus-Gaddum-type results for the convex domination number are studied.

Key words and phrases

convex domination number Nordhaus-Gaddum results 

Mathematics subject classification numbers

05C05 05C69 

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References

  1. [1]
    E. J. Cockayne and S. T. Hedetniemi, Total domination in graphs, Networks, 10 (1980), 211–219.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    J. Cyman, M. Lemańska and J. Raczek, Graphs with convex domination number close to their order, Discuss. Math. Graph Theory, 26 (2006), 307–316.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    G. S. Domke, J. H. Hattingh, S. T. Hedetniemi, R. C. Laskar and L. R. Markus, Restrained domination in graphs, Discrete Math., 203 (1999), 61–69.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    F. Harary and T. W. Haynes, Double domination in graphs, Ars Combin., 55 (2000), 201–213.MathSciNetMATHGoogle Scholar
  5. [5]
    F. Harary and T. W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs, Discrete Math., 155 (1996), 99–105.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., 1998.Google Scholar
  7. [7]
    S. T. Hedetniemi and R. Laskar, Connected domination in graphs, Graph Theory and Combinatorics (ed. B. Bollobás), Academic Press, London, 1984, 209–218.Google Scholar
  8. [8]
    Y. Hong and J. L. Shu, A sharp upper bound for the spectral radius of the Nordhaus-Gaddum type, Discrete Math., 211 (2000), 229–232.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    F. Jaegar, C. Payan, Relations du type Nordhaus-Gaddum pour le nombre d’absorption d’un graphe simple, C. R. Acad. Sci. Paris, 274 (1972), 728–730.Google Scholar
  10. [10]
    M. Lemańska, Weakly convex and convex domination numbers, Opuscula Math., 24 (2004), 181–188.MathSciNetMATHGoogle Scholar
  11. [11]
    M. Lemańska, Nordhaus-Gaddum results for the weakly convex domination number of a graph, Discuss. Math. Graph Theory, 30 (2010), 257–265.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    E. A. Nordhaus, J. W. Gaddum, On complementary graphs, Amer. Math. Monthly, 63 (1956), 175–177.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    J. Raczek and M. Lemańska, A note of the weakly convex and convex domination numbers of a torus, Discrete Appl. Math., 158 (2010), 1708–1713.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    J. Topp, Private communication, 2002.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  • M. Lemańska
    • 1
  • J. A. Rodríguez-Velázquez
    • 2
  • I. Gonzalez Yero
    • 2
  1. 1.Department of Technical Physics and Applied MathematicsGdańsk University of TechnologyGdańskPoland
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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