An end degree for digraphs

TL;DR

This paper introduces a new concept of degree for ends of infinite digraphs, resolving a previous problem and extending the notion to include dominating vertices, with a main result characterizing this degree via end-exhausting sequences.

Contribution

It defines a new end degree for infinite digraphs, resolves a known problem, and extends the concept to include dominating vertices, providing a characterization through end-exhausting sequences.

Findings

Defined a degree for ends of infinite digraphs.

Resolved a problem posed by Zuther.

Characterized the combined end degree using end-exhausting sequences.

Abstract

In this paper we define a degree for ends of infinite digraphs. The well-definedness of our definition in particular resolves a problem by Zuther. Furthermore, we extend our notion of end degree to also respect, among others, the vertices dominating the end, which we denote as combined end degree. Our main result is a characterisation of the combined end degree in terms of certain sequences of vertices, which we call end-exhausting sequences. This establishes a similar, although more complex relationship as known for the combined end degree and end-defining sequences in undirected graphs.