  Motivated by the work on the domination number of directed de Bruijn graphs
and some of its generalizations, in this paper we introduce a natural
generalization of de Bruijn graphs (directed and undirected), namely
$t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then
study the domination number of these graphs.
  Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and
Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs,
respectively. This generalization inherits many structural properties of de
Bruijn graphs and may have similar applications in interconnection networks or
bioinformatics.
  We establish upper and lower bounds for the domination number on
$t$-constrained de Bruijn graphs both in the directed and in the undirected
case. These bounds are often very close and in some cases we are able to find
the exact value.
