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

Author:

Thomas Robin

Abstract

We prove that given any sequence G 1 , G 2 , {G_1},{G_2}, \ldots of graphs, where G 1 {G_1} is finite planar and all other G i {G_i} are possibly infinite, there are indices i , j i,j such that i > j i > j and G i {G_i} is isomorphic to a minor of G j {G_j} . This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G 1 {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 QO {\text {QO}} -categories, which includes infinitary version of a well-quasi-ordering theorem of Friedman.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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