Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 Maker-Breaker total domination number 1249 1263 15177 10.22049/cco.2026.30824.2634 EN Athira Divakaran Department of Mathematics, Mar Athanasius College, Kothamangalam, India Tijo James Department of Mathematics, Pavanatma College, Murickassery, India Sandi Klavžar Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia Latha S Nair Department of Mathematics, Mar Athanasius College, Kothamangalam, India Journal Article 2025 07 23 The Maker-Breaker total domination number, $\gamma_{\rm MBT}(G)$, 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, $\gamma_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $\gamma_{\rm MBT}(G)$ and on $\gamma_{\rm MBT}'(G)$ 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 $\gamma_{\rm MB}(G)=k$ and $\gamma_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $\gamma_{\rm MB}'(G')=k$ and $\gamma_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $\gamma_{\rm MBT}(G'')=k$ and $\gamma_{\rm MBT}'(G'')=\ell$. Here, $\gamma_{\rm MB}$ and $\gamma_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game. https://comb-opt.azaruniv.ac.ir/article_15177_0972e39f2250a6f43cb2b34e6e32d563.pdf