| Title: | Fast winning strategies for Staller in the Maker-Breaker domination game |
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| Authors: | ID Bujtás, Csilla (Author)
ID Dokyeesun, Pakanun (Author) |
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| Files: |  PDF - Presentation file, download (503,82 KB)
MD5: 632DBA700A188F73A5DD64B3BCC8F945
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0166218X23004286
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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: | 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. |
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| Keywords: | domination game, Maker–Breaker game, winning number, Maker-Breaker domination game, closed neighborhood hypergraph |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.02.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 10-22 |
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| Numbering: | Vol. 344 |
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| PID: | 20.500.12556/DiRROS-20509  |
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| UDC: | 519.17 |
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| ISSN on article: | 0166-218X |
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| DOI: | 10.1016/j.dam.2023.11.015 |
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| COBISS.SI-ID: | 206680323 |
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| Note: | Spletna objava: 9. 11. 2023;
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| Publication date in DiRROS: | 03.10.2024 |
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| Views: | 211 |
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| Downloads: | 105 |
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