Maker-Breaker total domination number

The Maker-Breaker total domination number, γMBT(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, γMBT'(G), is defined analogously for the game in which Staller starts. Upper and lower bounds on γMBT(G) and on γMBT'(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 γMB(G)=k and γMBT(G)=, (ii) there exists a connected graph $G'$ with γMB'(G')=k and γMBT'(G')=, and (iii) there there exists a connected graph $G''$ with γMBT(G'')=k and γMBT'(G'')=. Here, γMB and γMB' are corresponding invariants for the Maker-Breaker domination game.