20.500.12556/DiRROS-23918 Thresholds for the biased Maker-Breaker domination games In the $(a,b)$-biased Maker-Breaker domination game, two players alternately select unplayed vertices in a graph $G$ such that Dominator selects $a$ and Staller selects $b$ vertices per move. Dominator wins if the vertices he selected during the game form a dominating set of $G$, while Staller wins if she can prevent Dominator from achieving this goal. Given a positive integer $b$, Dominator's threshold, ${\rm a}_b$, is the minimum $a$ such that Dominator wins the $(a,b)$-biased game on $G$ when he starts the game. Similarly, ${\rm a}'_b$ denotes the minimum $a$ such that Dominator wins when Staller starts the $(a,b)$-biased game. Staller's thresholds, ${\rm b}_a$ and ${\rm b}'_a$, are defined analogously. It is proved that Staller wins the $(k-1,k)$-biased games in a graph $G$ if its order is sufficiently large with respect to a function of $k$ and the maximum degree of $G$. Along the way, the $\ell$-local domination number of a graph is introduced. This new parameter is proved to bound Dominator's thresholds ${\rm a}_\ell$ and ${\rm a}_\ell'$ from above. As a consequence, ${\rm a}_1'(G)\le 2$ holds for every claw-free graph $G$. More specific results are obtained for thresholds in line graphs and Cartesian grids. Based on the concept of $[1,k]$-factor of a graph $G$, we introduce the star partition width $\sigma(G)$ of $G$, and prove that ${\rm a}_1'(G)\le \sigma(G)$ holds for any nontrivial graph $G$, while ${\rm a}_1'(G)=\sigma(G)$ if $G$ is a tree. Maker-Breaker domination game biased Maker-Breaker game trees line graph grid dominacijska igra izdelovalec-lomilec pristranska igra izdelovalec-lomilec drevesa graf povezav mreža true false true Angleški jezik Slovenski jezik Neznano 2025-10-23 12:51:18 2025-10-23 12:51:18 2025-10-24 03:55:18 0000-00-00 00:00:00 2025 0 0 19 str. no. 3, article no. 19 Vol. 27 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 1365-8050 10.46298/dmtcs.15392 254459907 2503.11871.pdf 2503.11871.pdf 1 BB9B0FBFFC769006FAC46A3270AB20E2 518c6b8f6a9b469ca7abac3fbdae086da4258d621944e59c6e8dabc3416708c0 9f64c4bb-affe-11f0-9bb7-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=35497 https://dmtcs.episciences.org/16768 1 36c23a78-affe-11f0-9bb7-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=35496 Inštitut za matematiko, fiziko in mehaniko 0 0 0