Ends of digraphs I: basic theory

Abstract: In a series of three papers we develop an end space theory for directed graphs. As for undirected graphs, the ends of a digraph are points at infinity to which its rays converge. Unlike for undirected graphs, some ends are joined by limit edges; these are crucial for obtaining the end space of a digraph as a natural (inverse) limit of its finite contraction minors. As our main result in this first paper of our series we show that the notion of directions of an undirected graph, a tangle-like description of its ends, extends to digraphs: there is a one-to-one correspondence between the `directions' of a digraph and its ends and limit edges. In the course of this we extend to digraphs a number of fundamental tools and techniques for the study of ends of graphs, such as the star-comb lemma and Schmidt's ranking of rayless graphs.