Parameterized Complexity of Generalized Domination Problems on Bounded Tree-Width Graphs

The concept of generalized domination unifies well-known variants of domination-like problems. A generalized domination (also called [Sigma,Rho]-domination) problem consists in finding a dominating set for which every vertex of the input graph is satisfied, given two sets of constraints Sigma and Rho. Very few problems are known to be W[1]-hard when restricted to graphs of bounded tree-width. We exhibit here a large new (infinite) collection of W[1]-hard problems parameterized by the tree-width of the input graph, that is [Sigma,Rho]-domination when Sigma is a set with arbitrary large gaps between two consecutive elements and Rho is cofinite (and an additional technical constraint on Sigma).