On computing graph minor obstruction sets

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 obstructions for the property. We prove several general theorems regarding the computation of obstruction 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 de ned on nite sets of graphs that generalizes the recognizability congruence that is de ned on single graphs. It is shown that the obstructions for a graph ideal can be e ectively 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 union F = F1 [ F2 of minor ideals can be computed from the obstruction sets for F1 and F2 if there is at least one tree that does not belong to the intersection of F1 and F2. As a corollary, it is shown that the set of intertwines of an arbitrary graph and a tree are e ectively computable.