Maker-Breaker total domination number

The Maker-Breaker total domination number, $γ_{\rm MBT}(G)$, 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, $γ_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $γ_{\rm MBT}(G)$ and on $γ_{\rm MBT}'(G)$ are provided and demonstrated to be sharp. It is proved that for any pair of integers $(k,\ell)$ with $2\leq k\leq \ell$, (i) there exists a connected graph $G$ with $γ_{\rm MB}(G)=k$ and $γ_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $γ_{\rm MB}'(G')=k$ and $γ_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $γ_{\rm MBT}(G'')=k$ and $γ_{\rm MBT}'(G'')=\ell$. Here, $γ_{\rm MB}$ and $γ_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game.