Abstract
We consider tournaments defined by several linear orderings of the vertex set according to some rule specifying the direction of each arc depending only on the order of the end vertices in each of the orderings. In the finite case, it was proved in Alon et al. (J Comb Theory Ser B 96:374–387, 2006) that the domination number of such tournaments is bounded in terms of the rule only. We show that for infinite tournaments, under some natural restrictions, the domination number is always finite, though in general cannot be bounded. We give some sufficient conditions under which there exists an upper bound in terms of the rule and/or the order types, and provide examples demonstrating that the bounds obtained are not very far from truth.
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Alon, N., Brightwell, G., Kierstead, H., Kostochka, A., Winkler, P.: Dominating sets in k-majority tournaments. J. Comb. Theory, Ser. B 96, 374–387 (2006)
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Partly supported by the grants 09-01-00244 and 08-01-00673 of the Russian Foundation for Fundamental Research.
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Fon-Der-Flaass, D.G. Domination Number of Certain Infinite Tournaments. Order 28, 157–161 (2011). https://doi.org/10.1007/s11083-010-9159-z
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DOI: https://doi.org/10.1007/s11083-010-9159-z