Fast winning strategies for Staller in the Maker-Breaker domination game

The Maker-Breaker domination game is played on a graph $G$ by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex $v \in V(G)$. Dominator's fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant $γ'_{\rm SMB}(G)$ (resp., $γ_{\rm SMB}(G)$) which is the smallest integer $k$ such that, under any strategy of Dominator, Staller can win the game by playing at most $k$ vertices, if Staller (resp., Dominator) plays first on the graph $G$. We prove some basic properties of $γ_{\rm SMB}(G)$ and $γ'_{\rm SMB}(G)$ and study the parameters' changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality $δ(G)+1 \le γ'_{\rm SMB}(G) \le γ_{\rm SMB}(G)$ always holds and that for every three integers $r,s,t$ with $2\le r\le s\le t$, there exists a graph $G$ such that $δ(G)+1 = r$, $γ'_{\rm SMB}(G) = s$, and $γ_{\rm SMB}(G) = t$. We prove exact formulas for $γ'_{\rm SMB}(G)$ where $G$ is a path, or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.

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