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On ubiquity problems in infinite digraphs

Matthias Hamann, Karl Heuer

Abstract

We prove that the consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs. Additionally, we prove the same equivalence for the disjoint union of a consistently forward and a consistently backward oriented ray. Furthermore, we discuss the connection between these two ubiquity problems.

On ubiquity problems in infinite digraphs

Abstract

We prove that the consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs. Additionally, we prove the same equivalence for the disjoint union of a consistently forward and a consistently backward oriented ray. Furthermore, we discuss the connection between these two ubiquity problems.
Paper Structure (5 sections, 13 theorems, 1 equation, 1 figure)

This paper contains 5 sections, 13 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the ubiquity for a pair of an anti-ray and a ray
  4. On the ubiquity for double rays
  5. On the equivalence of ubiquity problems

Key Result

Theorem 1.2

The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.

Figures (1)

  • Figure 2.1: The bidirected quarter-grid.

Theorems & Definitions (22)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more