Roughly speaking an end of a graph G(V,E) is an equivalence class of one-way infinite paths such that any two different equivalence classes can be separated from each other by removing a finite subset S of V(G). In this talk we consider infinite subsets of V(G) with certain growth rates which can be separated from each other by a connected subgraph of G with smaller growth rate. Following the ideas presented in a paper of Stallings we not only generalize the concept of ends, but also some of the well-known results about ends of graphs.