Dominating functions and topological graph minors

An infinite graph G is called dominating if its vertices can be labelled with natural numbers in such a way that for every function f : ω→ ω there is a ray in G whose sequence of labels eventually exceeds f . Conversely, G is called bounded if for every labelling of its vertices with natural numbers there exists a function f : ω→ω which eventually exceeds the labelling along any ray in G. This expository paper describes recent classifications of the dominating and the bounded graphs by forbidden topological minors, and indicates some connections of these results to infinite games.