On computing graph minor obstruction sets1

Communicated by M. Nivat The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a nite set of forbidden substructures, the obstruc-tions for the property. We prove several general theorems regarding the computation of obstruc-tion sets from other information about a family of graphs. The methods can be adapted to other partial orders on graphs, such as the immersion and topological orders. The algorithms are in some cases practical and have been implemented. Two new technical ideas are introduced. The rst is a method of computing a stopping signal for search spaces of increasing pathwidth. This allows obstruction sets to be computed without the necessity of a prior bound on maximum obstruction width. The second idea is that of a second order congruence for a graph property. This is an equivalence relation dened on nite sets of graphs that generalizes the recognizability congruence that is dened on single graphs. It is shown that the obstructions for a graph ideal can be eectively computed from an oracle for the canonical second-order congruence for the ideal and a membership oracle for the ideal. It is shown that the obstruction set for a unio