| Title: | Maker-Breaker domination game on trees when Staller wins |
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| Authors: | ID Bujtás, Csilla (Author)
ID Dokyeesun, Pakanun (Author)
ID Klavžar, Sandi (Author) |
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| Files: |  PDF - Presentation file, download (255,58 KB)
MD5: D9ABDC296CA33CAD76EB9FE4F838BFE7
 URL - Source URL, visit https://dmtcs.episciences.org/12202
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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 Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $\gamma_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $\gamma_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If ▫$n_1$▫ and $n_2$ are the two smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$▫. For caterpillars, exact formulas for $\gamma_{\rm SMB}$ and for $\gamma_{\rm SMB}'$ are established. |
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| Keywords: | domination game, Maker-Breaker game, Maker-Breaker domination game, hypergraphs, trees, subdivided stars, caterpillars |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2023 |
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| Year of publishing: | 2023 |
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| Number of pages: | 21 str. |
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| Numbering: | Vol. 25, no. 2, [article no.] 12 |
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| PID: | 20.500.12556/DiRROS-18631  |
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| UDC: | 519.17 |
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| ISSN on article: | 1365-8050 |
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| DOI: | 10.46298/dmtcs.10515 |
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| COBISS.SI-ID: | 164065283 |
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| Note: |
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| Publication date in DiRROS: | 08.04.2024 |
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| Views: | 633 |
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| Downloads: | 254 |
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| Metadata: |    |
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