Let H = (V, E) be a hypergraph with vertex set V and edge set E. A dominating set in H is a subset of vertices D ⊆ V such that for every vertex v ∈ V D there exists an edge e ∈ E for which v ∈ e and e ∩ D ≠ ∅. The domination number γ(H) is the minimum cardinality of a dominating set in H. It is known [Cs. Bujtás, M.A. Henning and Zs. Tuza, Transversals and domination in uniform hypergraphs, European J. Combin. 33 (2012) 62-71] that for k ≥ 5, if H is a hypergraph of order n and size m with all edges of size at least k and with no isolated vertex, then γ(H) ≤ (n + ⌊(k − 3)/2⌋m)/(⌊3(k − 1)/2⌋). In this paper, we apply a recent result of the authors on hypergraphs with large transversal number [M.A. Henning and C. Löwenstein, A characterization of hypergraphs that achieve equality in the Chvátal-McDiarmid Theorem, Discrete Math. 323 (2014) 69-75] to characterize the hypergraphs achieving equality in this bound.
@article{bwmeta1.element.doi-10_7151_dmgt_1865,
author = {Michael A. Henning and Christian L\"owenstein},
title = {A Characterization of Hypergraphs with Large Domination Number},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {427-438},
zbl = {1334.05102},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1865}
}
Michael A. Henning; Christian Löwenstein. A Characterization of Hypergraphs with Large Domination Number. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 427-438. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1865/