20.500.12556/RUL-163171 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 $\gamma'_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}(G)$) 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 $\gamma_{\rm SMB}(G)$ and $\gamma'_{\rm SMB}(G)$ 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 $\delta(G)+1 \le \gamma'_{\rm SMB}(G) \le \gamma_{\rm SMB}(G)$ 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$\delta(G)+1 = r$, $\gamma'_{\rm SMB}(G) = s$, and $\gamma_{\rm SMB}(G) = t$. We prove exact formulas for $\gamma'_{\rm SMB}(G)$ 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. domination game Maker–Breaker game winning number Maker-Breaker domination game closed neighborhood hypergraph true false true Angleški jezik Angleški jezik Članek v reviji 2024-10-03 09:56:44 2024-10-03 09:56:46 2024-10-08 03:32:23 0000-00-00 00:00:00 2024 0 0 Str. 10-22 Vol. 344 Feb. 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2024-02-01 519.17 0166-218X 10.1016/j.dam.2023.11.015 206680323 1-s2.0-S0166218X23004286-main-1.pdf 1-s2.0-S0166218X23004286-main-1.pdf 1 E43790D173D19824080A533754EF8171 0c6aa951b1da88e7cadb13732f965177f0c805144433bbfb0f2945539371d358 bccec104-815d-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=192690 https://www.sciencedirect.com/science/article/pii/S0166218X23004286 1 e19e6236-815c-11ef-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=192688 Fakulteta za matematiko in fiziko 0 0 0