  The concept of generalized domination unifies well-known variants of
domination-like and independence problems, such as Dominating Set, Independent
Set, Perfect Code, etc. A generalized domination (also called
$[\sigma,\rho]$-Dominating Set}) problem consists in finding a subset of
vertices in a graph such that every vertex is satisfied with respect to two
given sets of constraints $\sigma$ and $\rho$. Very few problems are known not
to be FPT when parameterized by tree-width, as usually this restriction allows
one to write efficient algorithms to solve the considered problems. The main
result of this article is a proof that for some (infinitely many) sets $\sigma$
and $\rho$, the problem $\exists[\sigma,\rho]$-Dominating Set} is W[1]-hard
when parameterized by the tree-width of the input graph. This contrasts with
the current knowledge on the parameterized complexity of this problem when
parameterized by tree-width, which had only been studied for finite and
cofinite sets $\sigma$ and $\rho$ and for which it has been shown to be FPT.
