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Transactions of the American Mathematical Society
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0932450-9
PII: S 0002-9947(1989)0932450-9
Keywords: Well-quasi-ordering, better-quasi-ordering, minor, infinite graph, Wagner's conjecture
Article copyright: © Copyright 1989 American Mathematical Society