Domination-like problems parameterized by tree-width
Résumé
The concept of generalized domination unifies well-known variants of domination-like problems. A generalized domination (also called [Sigma,Rho]-Dominating Set}) 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 not to be FPT when parameterized by tree-width, as usually this restriction allows to write efficient algorithms to solve the considered problems. We show in this paper 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. Up to now, its parameterized complexity had only been studied for finite and cofinite sets Sigma and Rho, for which it is FPT parameterized by tree-width.