Reinhard Diestel

Highly Connected Sets and the Excluded Grid Theorem

We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width. We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if and only if it contains no large highly connected set of vertices.

On Infinite Cycles I

We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S1 in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the end-faithful spanning trees. We also generalize Euler’s theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.

Graph-theoretical versus topological ends of graphs

We compare the notions of an end that exist in the graph-theoretical and, independently, in the topological literature. These notions conflict except for locally finite graphs, and we show how each can be expressed in the context of the other. We find that the topological ends of a graph are precisely the undominated of its graph-theoretical ends, and that graph theoretical ends have a simple topological description generalizing the definition of a topological end.

A Conjecture Concerning a Limit of Non-Cayley Graphs

Our aim in this note is to present a transitive graph that we conjecture is not quasi-isometric to any Cayley graph. No such graph is currently known. Our graph arises both as an abstract limit in a suitable space of graphs and in a concrete way as a subset of a product of trees.

Topological paths, cycles and spanning trees in infinite graphs

We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph and all its ends, but for one where only its topological ends are added as new points, while rays from other ends are made to converge to certain vertices.

The Cycle Space of an Infinite Graph

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new ‘singular’ approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of

Duality in Infinite Graphs

S^1

Two Short Proofs Concerning Tree-Decompositions

in the space formed by the graph together with its ends.Our approach permits the extension to infinite graphs of standard results about finite graph homology – such as cycle–cocycle duality and Whitneys theorem, Tuttes generating theorem, MacLanes planarity criterion, the Tutte/Nash-Williams tree packing theorem – whose infinite versions would otherwise fail. A notion of end degrees motivated by these results opens up new possibilities for an ‘extremal’ branch of infinite graph theory.Numerous open problems are suggested.

End spaces and spanning trees

The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassens results about ‘finitary’ duality for infinite graphs to full duality, including his extensions of Whitneys theorem.

The end structure of a graph: recent results and open problems

We give short proofs of the following two results: Thomass theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width; and the ‘tree-width duality theorem’ of Seymour and Thomas, that the tree-width of a finite graph is exactly one less than the largest order of its brambles.