Ends, tangles and critical vertex sets

TL;DR

This paper introduces a new way to compactify infinite graphs using ends and critical vertex sets, providing a unified framework that generalizes existing constructions and answers longstanding questions.

Contribution

It defines a new tangle compactification using critical vertex sets, compares it with Diestel's construction, and shows their relationship through inverse limits, addressing open questions.

Findings

The new compactification includes ends and critical vertex sets.

Both compactifications coincide if all tangles are ends.

The coarsest and finest compactifications are characterized by inverse limits.

Abstract

We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_{\Gamma}$ is a quotient of Diestel's (denoted by $\vert G\vert_{\Theta}$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of $\vert G\vert_{\Theta}$ and our construction of $\vert G\vert_{\Gamma}$, we show that $G$ can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's $\vert G\vert_{\Theta}$ is the finest such compactification, and our $\vert G\vert_{\Gamma}$ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.