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Graph minors. XVIII: Tree-decompositions and well-quasi-ordering. (English) Zbl 1023.05111
Summary: 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 Wagner’s conjecture.

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C05 Trees
05C65 Hypergraphs
05C83 Graph minors
Full Text: DOI
[1] C. St. J. A. Nash-Williams, On well-quasi-ordering trees, Theory of Graphs and its Applications (Proc. Symp. Smolenice, 1963), Publ. House Czechoslovak Acad. Sci., 1964, pp. 83-84.
[2] Robertson, Neil; Seymour, P.D., Graph minors. IV. tree-width and well-quasi-ordering, J. combin. theory ser. B, 48, 227-254, (1990) · Zbl 0719.05032
[3] Robertson, Neil; Seymour, P.D., Graph minors. X. obstructions to tree-decomposition, J. combin. theory ser. B, 52, 153-190, (1991) · Zbl 0764.05069
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