Advertisement

A Classification of Cactus Graphs According to Their Total Domination Number

  • Majid Hajian
  • Michael A. HenningEmail author
  • Nader Jafari Rad
Article
  • 26 Downloads

Abstract

A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The total domination number, \(\gamma _t(G)\), is the minimum cardinality of a total dominating set of G. A cactus is a connected graph in which every edge belongs to at most one cycle. Equivalently, a cactus is a connected graph in which every block is an edge or a cycle. Let G be a connected graph of order \(n \ge 2\) with \(k \ge 0\) cycles and \(\ell \) leaves. Recently, the authors have proved that \(\gamma _t(G) \ge \frac{1}{2}(n-\ell +2) - k\). As a consequence of this bound, \(\gamma _t(G) = \frac{1}{2}(n-\ell +2+m) - k\) for some integer \(m \ge 0\). In this paper, we characterize the class of cactus graphs achieving equality in this bound, thereby providing a classification of all cactus graphs according to their total domination number.

Keywords

Total dominating sets Total domination number Cactus graphs 

Mathematics Subject Classification

05C69 

Notes

References

  1. 1.
    Chellali, M., Haynes, T.W.: A note on the total domination number of a tree. J. Comb. Math. Comb. Comput. 58, 189–193 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cockayne, E.J., Henning, M.A., Mynhardt, C.M.: Vertices contained in all or in no minimum total dominating set of a tree. Discrete Math. 260, 37–44 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Desormeaux, W.J., Henning, M.A.: Lower bounds on the total domination number of a graph. J. Comb. Optim. 31(1), 52–66 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hajian, M., Henning, M.A., Jafari Rad, N.: A new lower bound on the total domination number of a graph (manuscript)Google Scholar
  5. 5.
    Henning, M.A.: Essential upper bounds on the total domination number. Discrete Appl. Math. 244, 103–115 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Henning, M.A., Yeo, A.: Total Domination in Graphs (Springer Monographs in Mathematics) (2013). ISBN: 978-1-4614-6524-9 (Print) 978-1-4614-6525-6Google Scholar
  7. 7.
    Henning, M.A., Yeo, A.: Total Domination in Trees. Chapter 3 in [7], pp. 19–29Google Scholar
  8. 8.
    Henning, M.A., Yeo, A.: Total Domination and Minimum Degree. Chapter 5 in [7], pp. 39–54Google Scholar
  9. 9.
    Henning, M.A., Yeo, A.: A new lower bound for the total domination number in graphs proving a Graffiti conjecture. Discrete Appl. Math. 173, 45–52 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  • Majid Hajian
    • 1
  • Michael A. Henning
    • 2
    Email author
  • Nader Jafari Rad
    • 3
  1. 1.Department of MathematicsShahrood University of TechnologyShahroodIran
  2. 2.Department of Mathematics and Applied MathematicsUniversity of JohannesburgJohannesburgSouth Africa
  3. 3.Department of MathematicsShahed UniversityTehranIran

Personalised recommendations