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

Abstract

Abstract.We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices ;', j such that i < j and G¡ is isomorphic to a minor of Gj . This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G\ 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 QOcategories, which includes infinitary version of a well-quasi-ordering theorem of Friedman.

All Graphs Have Tree-Decompositions Displaying Their Topological Ends

“…In [2,25,29], tree-decompositions of finite adhesion are used to study the 1 We denote the vertex set of a graph G by V (G). Figure 2: The ends of the decomposition tree -this tree is indicated in grey -define precisely the vertex-ends of the graph indicated by dots.…”

A short proof of Fleischner’s theorem

“…A well-known conjecture of Thomas [23] postulates that the countable graphs are well-quasi-ordered under the minor relation. The analogous statement for finite graphs is the celebrated Graph Minor Theorem of Robertson & Seymour [22].…”

“…A well-known conjecture of Thomas [23] postulates that the countable graphs are well-quasi-ordered under the minor relation. Our results suggest that the restriction of Thomas' conjecture to vertex-transitive graphs may be within reach.…”

Excluding a Countable Clique

“…First, K + 0 is in a sense the most general countable minor to exclude (as explained above for K n ) and is therefore a natural first choice. Second, there is the challenge to extend the Graph Minor Theorem to infinite graphs: it is known to be false in general [11], but was conjectured in [12] to extend to countable graphs. In a possible proof along the lines of the finite version, our K + 0 minor theorem might assume the role played there by the K n minor theorem: given a sequence G 0 , G 1 , .…”