Graphs and Combinatorics

, Volume 34, Issue 3, pp 415–425

# On the Game Total Domination Number

• Csilla Bujtás
Original Paper

## 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. 1.
Brešar, B., Klavžar, S., Rall, D.F.: Domination game and an imagination strategy. SIAM J. Discrete Math. 24, 979–991 (2010)
2. 2.
Brešar, B., Dorbec, P., Klavžar, S., Košmrlj, G.: Domination game: effect of edge- and vertex-removal. Discrete Math. 330, 1–10 (2014)
3. 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. 4.
Bujtás, Cs: Domination game on forests. Discrete Math. 338, 2220–2228 (2015)
5. 5.
Bujtás, Cs.: On the game domination number of graphs with given minimum degree. Electron. J. Combin. 22, #P3.29 (2015)Google Scholar
6. 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)
7. 7.
Bujtás, Cs, Henning, M.A., Tuza, Zs: Bounds on the game transversal number in hypergraphs. Eur. J. Combin. 59, 34–50 (2017)
8. 8.
Dorbec, P., Henning, M.A.: Game total domination for cycles and paths. Discrete Appl. Math. 208, 7–18 (2016)
9. 9.
Haynes, T.W., Hedetniemi, S.T., Slater, P.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)
10. 10.
Henning, M.A., Rall, D.F.: Progress towards the total domination game $$\frac{3}{4}$$-conjecture. Discrete Math. 339, 2620–2627 (2016)
11. 11.
Henning, M.A., Klavžar, S., Rall, D.F.: Total version of the domination game. Graphs Combin. 31, 1453–1462 (2015)
12. 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)
13. 13.
Kinnersley, W.B., West, D.B., Zamani, R.: Extremal problems for game domination number. SIAM J. Discrete Math. 27, 2090–2107 (2013)