# On the Game Total Domination Number

## Abstract

The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph *G*. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of *G*. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of *G*, \(\gamma _{\mathrm{tg}}(G)\), is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henning, Klavžar, and Rall proved that \(\gamma _{\mathrm{tg}}(G) \le \frac{4}{5}n\) holds for every graph *G* which is given on *n* vertices such that every component of it is of order at least 3; they also conjectured that the sharp upper bound would be \(\frac{3}{4}n\). Here, we prove that \(\gamma _{\mathrm{tg}}(G) \le \frac{11}{14}n\) holds for every *G* which contains no isolated vertices or isolated edges.

## Keywords

Dominating set Total dominating set Total domination game Open neighborhood hypergraph Transversal game## Mathematics Subject Classification

05C69 05C65 05C57## References

- 1.Brešar, B., Klavžar, S., Rall, D.F.: Domination game and an imagination strategy. SIAM J. Discrete Math.
**24**, 979–991 (2010)MathSciNetCrossRefMATHGoogle Scholar - 2.Brešar, B., Dorbec, P., Klavžar, S., Košmrlj, G.: Domination game: effect of edge- and vertex-removal. Discrete Math.
**330**, 1–10 (2014)MathSciNetCrossRefMATHGoogle Scholar - 3.Bujtás, Cs.: Domination game on trees without leaves at distance four. In: Frank, A., Recski, A., Wiener, G. (eds.) Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, Veszprém, pp. 73–78 (2013)Google Scholar
- 4.Bujtás, Cs: Domination game on forests. Discrete Math.
**338**, 2220–2228 (2015)MathSciNetCrossRefMATHGoogle Scholar - 5.Bujtás, Cs.: On the game domination number of graphs with given minimum degree. Electron. J. Combin.
**22**, #P3.29 (2015)Google Scholar - 6.Bujtás, Cs, Henning, M.A., Tuza, Zs: Transversal game on hypergraphs and the \(\frac{3}{4}\)-conjecture on the total domination game. SIAM J. Discrete Math.
**30**, 1830–1847 (2016)MathSciNetCrossRefMATHGoogle Scholar - 7.Bujtás, Cs, Henning, M.A., Tuza, Zs: Bounds on the game transversal number in hypergraphs. Eur. J. Combin.
**59**, 34–50 (2017)MathSciNetCrossRefMATHGoogle Scholar - 8.Dorbec, P., Henning, M.A.: Game total domination for cycles and paths. Discrete Appl. Math.
**208**, 7–18 (2016)MathSciNetCrossRefMATHGoogle Scholar - 9.Haynes, T.W., Hedetniemi, S.T., Slater, P.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)MATHGoogle Scholar
- 10.Henning, M.A., Rall, D.F.: Progress towards the total domination game \(\frac{3}{4}\)-conjecture. Discrete Math.
**339**, 2620–2627 (2016)MathSciNetCrossRefMATHGoogle Scholar - 11.Henning, M.A., Klavžar, S., Rall, D.F.: Total version of the domination game. Graphs Combin.
**31**, 1453–1462 (2015)MathSciNetCrossRefMATHGoogle Scholar - 12.Henning, M.A., Klavžar, S., Rall, D.F.: The 4/5 upper bound on the game total domination number. Combinatorica
**31**, 223–251 (2017)MathSciNetCrossRefMATHGoogle Scholar - 13.Kinnersley, W.B., West, D.B., Zamani, R.: Extremal problems for game domination number. SIAM J. Discrete Math.
**27**, 2090–2107 (2013)MathSciNetCrossRefMATHGoogle Scholar