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An end degree for digraphs

Matthias Hamann, Karl Heuer

TL;DR

Addresses extending end-degree concepts to ends of infinite digraphs and resolves Zuther's problem by defining the combined end degree via dominating vertices. Develops end-exhausting sequences and proves that for a countable end with at least one ray, the combined in-degree satisfies $K(\omega)=\Delta^-(\omega)$, with $K(\omega):=\inf\{\liminf_{i\in\mathbb N}|U_i| : (U_i)\text{ is an }\omega\text{-exhausting sequence}\}$, and analogously for the out-degree. The paper also extends these results to edge-disjoint structures and discusses implications for the broader structure of ends in infinite digraphs. Overall, it builds a robust, sequence-based framework that parallels the undirected case and broadens our understanding of directed end-structure and domination phenomena.

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.

An end degree for digraphs

TL;DR

Addresses extending end-degree concepts to ends of infinite digraphs and resolves Zuther's problem by defining the combined end degree via dominating vertices. Develops end-exhausting sequences and proves that for a countable end with at least one ray, the combined in-degree satisfies , with , and analogously for the out-degree. The paper also extends these results to edge-disjoint structures and discusses implications for the broader structure of ends in infinite digraphs. Overall, it builds a robust, sequence-based framework that parallels the undirected case and broadens our understanding of directed end-structure and domination phenomena.

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. End degree
  4. Example
  5. End-exhausting sequences
  6. Edge-degrees of ends

Key Result

Theorem 1.1

Let $D$ be a digraph and let $\omega$ be an end of $D$ that contains at least one but at most countably many rays. Then the combined in-degree of $\omega$ is the same as where both values are considered in $\mathbb N\cup\{\infty\}$.

Figures (2)

  • Figure 4.1: Two drawings of the digraph $D$.
  • Figure 5.1: The end containing $R$ has combined in-degree $2$, but contains no two disjoint rays and is not dominated by any vertex.

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1: Star-Comb Lemma
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Proposition 4.4
  • proof
  • Proposition 5.1
  • ...and 15 more