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

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.