# On the Game Total Domination Number

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## 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

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