Maker–Breaker Resolving Game

Kang, Cong X.; Klavžar, Sandi; Yero, Ismael G.; Yi, Eunjeong

doi:10.1007/s40840-020-01044-0

Cited by 6 publications

(1 citation statement)

References 20 publications

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“…The game, either in its general form, or in different special cases, was investigated a lot by now, see the book Hefetz et al (2014). For related recent developments see Clemens et al (2021); ; Nicholas ; Glazik and Srivastav (2022); Kang et al (2021); Stojaković and Trkulja (2021) and references therein. The Maker-Breaker games have been recently studied also on digraphs in Frieze and Pegden (2021).…”

Section: Introductionmentioning

confidence: 99%

Maker-Breaker domination game on trees when Staller wins

Bujtás,

Dokyeesun,

Klavžar

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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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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Section: Introductionmentioning

confidence: 99%

Maker-Breaker domination game on trees when Staller wins

Bujtás,

Dokyeesun,

Klavžar

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Fractional Maker-Breaker Resolving Game

2020

Combinatorial Optimization and Applications

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Maker-Breaker resolving game played on corona products of graphs

James,

Klavžar,

Kuziak

et al. 2024

Aequat. Math.

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The Maker-Breaker resolving game is a game played on a graph G by Resolver and Spoiler. The players taking turns alternately in which each player selects a not yet played vertex of G. The goal of Resolver is to select all the vertices in a resolving set of G, while that of Spoiler is to prevent this from happening. The outcome o(G) of the game played is one of $$\mathcal {R}$$ R , $$\mathcal {S}$$ S , and $$\mathcal {N}$$ N , where $$o(G)=\mathcal {R}$$ o ( G ) = R (resp. $$o(G)=\mathcal {S}$$ o ( G ) = S ), if Resolver (resp. Spoiler) has a winning strategy no matter who starts the game, and $$o(G)=\mathcal {N}$$ o ( G ) = N , if the first player has a winning strategy. In this paper, the game is investigated on corona products $$G\odot H$$ G ⊙ H of graphs G and H. It is proved that if $$o(H)\in \{\mathcal {N}, \mathcal {S}\}$$ o ( H ) ∈ { N , S } , then $$o(G\odot H) = \mathcal {S}$$ o ( G ⊙ H ) = S . No such result is possible under the assumption $$o(H) = \mathcal {R}$$ o ( H ) = R . It is proved that $$o(G\odot P_k) = \mathcal {S}$$ o ( G ⊙ P k ) = S if $$k=5$$ k = 5 , otherwise $$o(G\odot P_k) = \mathcal {R}$$ o ( G ⊙ P k ) = R , and that $$o(G\odot C_k) = \mathcal {S}$$ o ( G ⊙ C k ) = S if $$k=3$$ k = 3 , otherwise $$o(G\odot C_k) = \mathcal {R}$$ o ( G ⊙ C k ) = R . Several results are also given on corona products in which the second factor is of diameter at most 2.

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Maker–Breaker Resolving Game

Cited by 6 publications

References 20 publications

Maker-Breaker domination game on trees when Staller wins

Maker-Breaker domination game on trees when Staller wins

Fractional Maker-Breaker Resolving Game

Maker-Breaker resolving game played on corona products of graphs

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