Associating constraints with nodes possibly reduces the width of decompositions and of instances. On the other hand, this definition of width still allows problems of fixed width to be solved in polynomial time if the decomposition is given. In this case, the domain of a new variable is obtained by solving a subproblem which can be polynomially large but has a fixed number of constraints. As a result, this domain is guaranteed to be of polynomial size; the constraints of the new problem, being equalities of two domains, are polynomial in size as well.