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 $[σ,ρ]$-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 $σ$ and $ρ$. 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 $σ$ and $ρ$, the problem $\exists[σ,ρ]$-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 $σ$ and $ρ$ and for which it has been shown to be FPT.

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