Maker-Breaker total domination number

Abstract

The Maker-Breaker total domination number, ${\hat{I}}_{\mathrm{MBT}}^3(G)$, 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 ${\hat{I}}_{\mathrm{MBT}}^3(G)$ 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 $2\le k\le \ell$, (i) there exists a connected graph $G$ with ${\hat{I}}_{\mathrm{MB}}^3(G)=k$ and ${\hat{I}}_{\mathrm{MBT}}^3(G)=\ell$, (ii) there exists a connected graph $G^{\prime }$ with $γ_{\rm MB}'(G')=k$ and $γ_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G^{\prime \prime }$ with ${\hat{I}}_{\mathrm{MBT}}^3(G^{\prime \prime })=k$ and $γ_{\rm MBT}'(G'')=\ell$. Here, ${\hat{I}}_{\mathrm{MB}}^3$ and $γ_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game.