Unavoidable topological minors of infinite graphs

A graph G is loosely- c -connected, or ℓ - c -connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of ℓ - c -connected infinite graphs that form an unavoidable set among the topological minors of ℓ - c -connected infinite graphs. Corresponding results for minors and parallel minors are also obtained.