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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Well-quasi-ordering infinite graphs with forbidden finite planar minor

Author: Robin Thomas
Journal: Trans. Amer. Math. Soc. 312 (1989), 279-313
MSC: Primary 05C99; Secondary 04A20
MathSciNet review: 932450
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Abstract: We prove that given any sequence $ {G_1},{G_2}, \ldots $ of graphs, where $ {G_1}$ is finite planar and all other $ {G_i}$ are possibly infinite, there are indices $ i,j$ such that $ i < j$ and $ {G_i}$ is isomorphic to a minor of $ {G_j}$ . This generalizes results of Robertson and Seymour to infinite graphs. The restriction on $ {G_1}$ cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem of Robertson and Seymour, its extension to infinite graphs, Nash-Williams' theory of better-quasi-ordering, especially his infinite tree theorem, and its extension to something we call tree-structures over $ {\text{QO}}$-categories, which includes infinitary version of a well-quasi-ordering theorem of Friedman.

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Keywords: Well-quasi-ordering, better-quasi-ordering, minor, infinite graph, Wagner's conjecture
Article copyright: © Copyright 1989 American Mathematical Society

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