Abstract

The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a finite set of forbidden substructures, the {\it 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 first is a method of computing a stopping signal for search spaces of expanding pathwidth. This allows obstruction sets to be computed for the first time without the necessity of a prior bound on maximum obstruction width. The second idea is that of a {\it second order congruence} for a graph property. This is an equivalence relation defined on finite sets of graphs that generalizes the recognizability congruence that is defined on single graphs. It is shown that the obstructions for a graph ideal can be effectively 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 ${\cal F}= {\cal F}{1} \cup {\cal F}{2}$ of minor ideals can be computed from the obstruction sets for ${\cal F}{1}$ and ${\cal F}{2}$ if there is at least one tree that does not belong to the intersection of ${\cal F}{1}$ and ${\cal F}{2}$. As a corollary, it is shown that the set of intertwines of an arbitrary graph and a tree are effectively computable.