Graphs and Combinatorics

, Volume 25, Issue 1, pp 41–47 | Cite as

A New Bound on the Total Domination Subdivision Number

  • O. Favaron
  • H. Karami
  • R. Khoeilar
  • S. M. Sheikholeslami


A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number \(sd_{\gamma_{t}}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for every simple connected graph G of order n ≥ 3,
$${\rm sd}_{\gamma_{t}}(G)\le 3 +{\rm min}\{d_2(v); v\in V \, {\rm and}\, d(v)\ge 2\}$$
where d 2(v) is the number of vertices of G at distance 2 from v.


Total domination number Total domination subdivision number 


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  1. 1.
    Bhattacharya, A., Vijayakumar, G.R.: Effect of edge-subdivision on vertex-domination in a graph. Discuss. Math. Graph Theory 22, 335–347 (2002)Google Scholar
  2. 2.
    Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks 10, 211–219 (1980)Google Scholar
  3. 3.
    Erdős, P., Rényi, A.: On a problem in the theory of graphs. Magyar Tud. Akad. Mat. Kutató Int. Kőzl. 7, 623–641 (1962)Google Scholar
  4. 4.
    Favaron, O., Karami, H., Khoeilar, R., Sheikholeslami, S.M.: On the total subdivision number in some classes of graphs, J. Comb. Optim. (to appear)Google Scholar
  5. 5.
    Favaron, O., Karami, H., Sheikholeslami, S.M.: Total domination and total domination subdivision numbers. Australas. J. Combin. 38, 229–235, (2007)Google Scholar
  6. 6.
    Favaron, O., Karami, H., Sheikholeslami, S.M.: Bounding the total domination subdivision number of a graph in terms of its order (submitted)Google Scholar
  7. 7.
    Haynes, T.W., Hedetniemi, S.T., van der Merwe, L.C.: Total domination subdivision numbers. J. Combin. Math. Combin. Comput. 44, 115–128 (2003)Google Scholar
  8. 8.
    Haynes, T.W., Henning, M.A., Hopkins, L.S.: Total domination subdivision numbers of graphs, Discuss. Math. Graph Theory 24, 457–467, (2004)Google Scholar
  9. 9.
    Karami, H., Khodkar, A., Sheikholeslami, S.M.: An upper bound for total domination subdivision numbers of graphs. Ars Combin. (to appear)Google Scholar
  10. 10.
    Velammal, S.: Studies in Graph Theory: Covering, Independence, Dominationand Related Topics, Ph.D. Thesis (Manonmaniam Sundaranar University, Tirunelveli, 1997)Google Scholar
  11. 11.
    West, D.B.: Introduction to Graph Theory (Prentice-Hall, Inc, 2000)Google Scholar

Copyright information

© Springer Japan 2009

Authors and Affiliations

  • O. Favaron
    • 1
  • H. Karami
    • 2
  • R. Khoeilar
    • 2
  • S. M. Sheikholeslami
    • 2
  1. 1.Univ Paris-Sud, LRI, UMR 8623OrsayFrance
  2. 2.Department of MathematicsAzarbaijan University of Tarbiat MoallemTabrizI.R. Iran

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