On the number of k-dominating independent sets

We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of $k$-dominating independent sets in $n$-vertex graphs is between $c_k\cdot {\sqrt[2k]{2}}^n$ and $c_k^{\mathrm{\prime }}\cdot {\sqrt[k+1]{2}}^n$ if $k\ge 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot {1.22}^n$ and $c^{\mathrm{\prime }}\cdot {1.246}^n$. Graph constructions containing a large number of $k$-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.