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The domination game played on diameter 2 graphs

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Abstract

Let \(\gamma _g(G)\) be the game domination number of a graph G. It is proved that if \(\mathrm{diam}(G) = 2\), then \(\gamma _g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil - \left\lfloor \frac{n(G)}{11}\right\rfloor \). The bound is attained: if \(\mathrm{diam}(G) = 2\) and \(n(G) \le 10\), then \(\gamma _g(G) = \left\lceil \frac{n(G)}{2} \right\rceil \) if and only if G is one of seven sporadic graphs with \(n(G)\le 6\) or the Petersen graph, and there are exactly ten graphs of diameter 2 and order 11 that attain the bound.

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Acknowledgements

We are grateful to Gašper Košmrlj for providing us with his software that computes game domination invariants. We acknowledge the financial support from the Slovenian Research Agency (research core Funding No. P1-0297 and Projects J1-9109, J1-1693, N1-0095, N1-0108). Kexiang Xu is also supported by NNSF of China (Grant No. 11671202) and China-Slovene bilateral Grant 12-9.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Csilla Bujtás, Vesna Iršič & Sandi Klavžar

  2. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Vesna Iršič & Sandi Klavžar

  3. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

  4. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, People’s Republic of China

    Kexiang Xu

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  1. Csilla Bujtás
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  2. Vesna Iršič
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  4. Kexiang Xu
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Correspondence to Vesna Iršič.

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Bujtás, C., Iršič, V., Klavžar, S. et al. The domination game played on diameter 2 graphs. Aequat. Math. 96, 187–199 (2022). https://doi.org/10.1007/s00010-021-00786-x

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  • DOI: https://doi.org/10.1007/s00010-021-00786-x

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