# On Upper Total Domination Versus Upper Domination in Graphs

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## Abstract

A total dominating set of a graph *G* is a dominating set *S* of *G* such that the subgraph induced by *S* contains no isolated vertex, where a dominating set of *G* is a set of vertices of *G* such that each vertex in \(V(G){\setminus } S\) has a neighbor in *S*. A (total) dominating set *S* is said to be minimal if \(S{\setminus } \{v\}\) is not a (total) dominating set for every \(v\in S\). The upper total domination number \(\varGamma _t(G)\) and the upper domination number \(\varGamma (G)\) are the maximum cardinalities of a minimal total dominating set and a minimal dominating set of *G*, respectively. For every graph *G* without isolated vertices, it is known that \(\varGamma _t(G)\le 2\varGamma (G)\). The case in which \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\) has been studied in Cyman et al. (Graphs Comb 34:261–276, 2018), which focused on the characterization of the connected cubic graphs and proposed one problem to be solved and two questions to be answered in terms of the value of \(\frac{\varGamma _t(G)}{\varGamma (G)}\). In this paper, we solve this problem, i.e., the characterization of the subcubic graphs *G* that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\), by constructing a class of subcubic graphs, which we call triangle-trees. Moreover, we show that the answers to the two questions are negative by constructing connected cubic graphs *G* that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}>\frac{3}{2}\) and a class of regular non-complete graphs *G* that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\).

## Keywords

Upper domination number Upper total domination number Subcubic graphs## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (61872101, 61672051, 61702075), the China Postdoctoral Science Foundation under Grant (2017M611223).

## References

- 1.Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
- 2.Brešar, B., Henning, M.A., Rall, D.F.: Total domination in graphs. Discrete Math.
**339**, 1665–1676 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Chen, J., He, K., Du, R., Zheng, M., Xiang, Y., Yuan, Q.: Dominating set and network coding-based routing in wireless mesh networks. IEEE Trans. Parallel Distrib. Syst.
**26**(2), 423–433 (2015)CrossRefGoogle Scholar - 4.Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks
**10**, 211–219 (1980)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Cyman, J., Dettlaff, M., Henning, M.A., Lemańska, M., Raczek, J.: Total domination versus domination in cubic graphs. Graphs Comb.
**34**, 261–276 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Desormeaux, W.J., Haynes, T.W., Henning, M.A., Yeo, A.: Total domination numbers of graphs with diameter two. J. Graph Theory
**75**, 91–103 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Dorbec, P., Henning, M.A., Rall, D.F.: On the upper total domination number of cartesian products of graphs. J. Comb. Optim.
**16**, 68–80 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP Completeness. W H Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
- 9.Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
- 10.Henning, M.A.: A survey of selected recent results on total domination in graphs. Discrete Math.
**309**(1), 32–63 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 11.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 - 12.Molnár, F., Sreenivasan, S., Szymanski, B.K., Korniss, G.: Minimum dominating sets in scale-free network ensembles. Sci. Rep.
**3**, 1736 (2013)CrossRefGoogle Scholar - 13.Wuchty, S.: Controllability in protein interaction networks. Proc. Natl. Acad. Sci. USA PNAS
**111**(19), 7156–7160 (2014)CrossRefGoogle Scholar