Criticality for Maker-Breaker domination games with predomination

A predominated graph is a pair $(G,D)$, where $G$ is a graph and the vertices in $D\subseteq V(G)$ are considered already dominated. Maker-Breaker domination game critical (MBD critical) predominated graphs are introduced as the predominated graphs $(G,D)$ on which Staller wins the game, but Dominator wins on $(G, D \cup \{v\})$ for every vertex $v \in V(G) \setminus D$. Tools are developed for handling the Maker-Breaker domination game on trees which lead to a characterization of Staller-win predominated trees. MBD critical predominated trees are characterized and an algorithm is designed which verifies in linear time whether a given predominated tree is MBD critical. A large class of MBD critical predominated cacti is presented and Maker-Breaker critical hypergraphs constructed.