Maker-Breaker total domination number

The Maker-Breaker total domination number, γ ₌₁ₓ (G), of a graph G is introduced as the minimum number of moves of Dominator to win the Maker-Breaker total domination game, provided that he has a winning strategy and is the first to play. The Staller-start Maker-Breaker total domination number, γ ₌₁ₓ' (G), is defined analogously for the game in which Staller starts. Upper and lower bounds on γ ₌₁ₓ (G) and on γ ₌₁ₓ' (G) are provided and demonstrated to be sharp. It is proved that for any pair of integers (k, ) with 2 k, (i) there exists a connected graph G with γ ₌₁ (G) =k and γ ₌₁ₓ (G) =, (ii) there exists a connected graph G' with γ ₌₁' (G') =k and γ ₌₁ₓ' (G') =, and (iii) there there exists a connected graph G'' with γ ₌₁ₓ (G'') =k and γ ₌₁ₓ' (G'') =. Here, γ ₌₁ and γ ₌₁' are corresponding invariants for the Maker-Breaker domination game.