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The concept of generalized domination unifies well-known variants ofdomination-like and independence problems, such as Dominating Set, IndependentSet, Perfect Code, etc. A generalized domination (also called$[\sigma,\rho]$-Dominating Set}) problem consists in finding a subset ofvertices in a graph such that every vertex is satisfied with respect to twogiven sets of constraints $\sigma$ and $\rho$. Very few problems are known notto be FPT when parameterized by tree-width, as usually this restriction allowsone to write efficient algorithms to solve the considered problems. The mainresult 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]-hardwhen parameterized by the tree-width of the input graph. This contrasts withthe current knowledge on the parameterized complexity of this problem whenparameterized by tree-width, which had only been studied for finite andcofinite sets $\sigma$ and $\rho$ and for which it has been shown to be FPT.

W[1]-hardness of some domination-like problems parameterized by tree-width