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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.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C05 Trees 05C65 Hypergraphs 05C83 Graph minors
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##### References:
 [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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