Connectoids II: existence of normal trees

TL;DR

This paper generalizes the concept of normal trees to connectoids, unifying connectivity structures across various discrete objects and establishing conditions for their existence.

Contribution

It introduces normal trees of connectoids, extending classical graph theory tools to a broader class of connectivity structures.

Findings

Normal trees of connectoids can be characterized similarly to those in undirected graphs.

Existence of normal spanning trees depends on the connectoid's neighborhood structure.

Normal spanning trees exist if the connectoid's groundset has a countable separation number.

Abstract

In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids. In the first paper, we developed a universal end space theory based on connectoids that unifies the existing end spaces of undirected and directed graphs. In this paper, we establish normal trees of connectoids as a natural generalisation of normal trees of undirected graphs, which are one of the most important tools in infinite graph theory. More precisely, we show that the existence of normal trees of connectoids can be characterised in the same way as for normal trees of undirected graphs: We extend Jung's famous characterisation via dispersed sets to connectoids, and prove that normal spanning trees exist if they exist in some neighbourhood of each end. Furthermore, we show that a connectoid has a normal spanning tree if and only if its groundset can be well-ordered in a certain way, called countable separation number.