| Title: | Thresholds for the biased Maker-Breaker domination games |
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| Authors: | ID Brešar, Boštjan (Author)
ID Bujtás, Csilla (Author)
ID Dokyeesun, Pakanun (Author)
ID Dravec, Tanja (Author) |
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| Files: |  PDF - Presentation file, download (415,24 KB)
MD5: BB9B0FBFFC769006FAC46A3270AB20E2
 URL - Source URL, visit https://dmtcs.episciences.org/16768
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | Maker-Breaker domination game, biased Maker-Breaker game, trees, line graph, grid |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 19 str. |
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| Numbering: | Vol. 27, no. 3, article no. 19 |
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| PID: | 20.500.12556/DiRROS-23918  |
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| UDC: | 519.17 |
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| ISSN on article: | 1365-8050 |
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| DOI: | 10.46298/dmtcs.15392 |
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| COBISS.SI-ID: | 254459907 |
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| Note: |
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| Publication date in DiRROS: | 23.10.2025 |
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| Views: | 172 |
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| Downloads: | 110 |
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| Metadata: |    |
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