Bounds on the Maximum Number of Minimum Dominating Sets
Given a graph with domination number γ, we find bounds on the maximum number of minimum dominating sets. First, for γ≥3, we obtain lower bounds on the number of γ-sets that do not dominate a graph on n vertices. Then, we show that γ-fold lexicographic product of the complete graph on n1/γ vertices has domination number γ and γn-O(nγ-γ/1) dominating sets of size γ. Finally, we see that a certain random graph has, with high probability, (i) domination number γ; and (ii) all but o(nγ) of its γ-sets being dominating.
Connolly, Samuel; Gabor, Zachary; Godbole, Anant; Kay, Bill; and Kelly, Thomas. 2016. Bounds on the Maximum Number of Minimum Dominating Sets. Discrete Mathematics. Vol.339(5). 1537-1542. https://doi.org/10.1016/j.disc.2015.12.030 ISSN: 0012-365X