Recent results of Robertson and Seymour show that every class that is closed under taking of minors can be recognized in (n³) time. If there is a fixed upper bound on the treewidth of the graphs in the class, i.e., if there is a planar graph not in the class, then the class can be recognized in (n²) time. However, this result is nonconstructive in two ways: the algorithm only decides on membership, but does not construct “a solution”, e.g., a linear ordering, decomposition or embedding; and no method is given to find the algorithms. In many cases, both nonconstructive elements can be avoided, using techniques of Brown (1989) and Fellows and Langston (1989), based on self-reduction. In this paper we introduce two techniques that help to reduce the running time of self-reduction algorithms. With the help of these techniques we show that there exist (n²) algorithms that decide on membership and construct solutions for treewidth, pathwidth, search number, vertex search number, node search number, cutwidth, modified cutwidth, vertex separation number, gate matrix layout, and progressive black–white pebbling, where in each case the parameter k is a fixed constant.