# A Classification of Cactus Graphs According to Their Total Domination Number

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## 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.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.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.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.Hajian, M., Henning, M.A., Jafari Rad, N.: A new lower bound on the total domination number of a graph (manuscript)Google Scholar
- 5.Henning, M.A.: Essential upper bounds on the total domination number. Discrete Appl. Math.
**244**, 103–115 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Henning, M.A., Yeo, A.: Total Domination in Trees. Chapter 3 in [7], pp. 19–29Google Scholar
- 8.Henning, M.A., Yeo, A.: Total Domination and Minimum Degree. Chapter 5 in [7], pp. 39–54Google Scholar
- 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