Ends of digraphs I: basic theory

TL;DR

This paper develops a foundational theory of ends for directed graphs, extending concepts from undirected graphs and introducing the notion of directions, limit edges, and their correspondence with ends.

Contribution

It introduces a new end space theory for digraphs, including the concept of directions and their relation to ends and limit edges, extending existing tools from undirected graph theory.

Findings

Establishes a one-to-one correspondence between directions and ends/limit edges in digraphs.

Extends fundamental tools like the star-comb lemma to directed graphs.

Shows that the end space of a digraph can be described as an inverse limit of its minors.

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.