Connectoids I: a universal end space theory

TL;DR

This paper introduces connectoids, a unifying framework for the connectivity and end space theory of various discrete structures, extending classical concepts from graphs to more general objects like hypergraphs and matroids.

Contribution

It develops a universal end space theory for connectoids, unifying and extending existing end space concepts across multiple discrete structures.

Findings

Connectoids unify end spaces of undirected and directed graphs.

End spaces of connectoids correspond to their directions.

Normal trees represent ends of connectoids.

Abstract

In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids. In this paper we develop a universal end space theory based on connectoids: the end spaces of connectoids unify the existing end spaces of undirected and directed graphs and establish end spaces for bidirected graphs, hypergraphs and finitary matroids. The main result shows that the tangle-like description of ends in undirected graphs, called directions, extends to connectoids: there is a one-to-one correspondence between the directions of a connectoid and its ends. Furthermore, we generalise normal trees of undirected graphs to connectoids and show that normal trees represent the ends of a connectoid as they do for undirected graphs.