  Erd\H{o}s and Moser raised the question of determining the maximum number of
maximal cliques or equivalently, the maximum number of maximal independent sets
in a graph on $n$ vertices. Since then there has been a lot of research along
these lines.
  A $k$-dominating independent set is an independent set $D$ such that every
vertex not contained in $D$ has at least $k$ neighbours in $D$. Let $mi_k(n)$
denote the maximum number of $k$-dominating independent sets in a graph on $n$
vertices, and let $\zeta_k:=\lim_{n \rightarrow \infty} \sqrt[n]{mi_k(n)}$.
Nagy initiated the study of $mi_k(n)$.
  In this article we disprove a conjecture of Nagy and prove that for any even
$k$ we have $$1.489 \approx \sqrt[9]{36} \le \zeta^k_k.$$
  We also prove that for any $k \ge 3$ we have $$\zeta_k^{k} \le
2.053^{\frac{1}{1.053+1/k}}< 1.98,$$ improving the upper bound of Nagy.
