Graph Minors. XVIII. Tree-decompositions and well-quasi-ordering

We prove the following result. Suppose that for every graph G in a class C of graphs, and for every “highly connected component” of G, there is a decomposition of G of a certain kind centred on the component. Then C is well-quasi-ordered by minors; that is, in any infinite subset of C there are two graphs, one a minor of the other. This is another step towards Wagnerʹs conjecture.