Unavoidable topological minors of infinite graphs

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