On ubiquity problems in infinite digraphs

TL;DR

This paper investigates the conditions under which certain infinite directed graphs, specifically oriented double rays and unions of oriented rays, are ubiquitous, establishing equivalences and exploring their interconnections.

Contribution

It proves the equivalence of ubiquity for specific infinite digraphs across different classes and discusses their relationship, advancing understanding of ubiquity in infinite digraphs.

Findings

Consistently oriented double ray is ubiquitous iff it is in one-ended digraphs.

Union of forward and backward oriented rays is ubiquitous iff in one-ended digraphs.

Explores the connection between two key ubiquity problems.

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.