Ends and Tangles

TL;DR

This paper generalizes the concept of ends in locally finite graphs by introducing ${\aleph_0}$-tangles, providing a new compactification method that encompasses all infinite graphs and relates to ultrafilter limits.

Contribution

It extends the theory of graph ends to arbitrary infinite graphs using ${\aleph_0}$-tangles and describes their structure via inverse limits of ultrafilters.

Findings

${\\aleph_0}$-tangles form an inverse limit of ultrafilters.

Ends are exactly the limits of principal ultrafilters.

Highly connected parts correspond to closed ${\\aleph_0}$-tangles.

Abstract

We show that an arbitrary infinite graph can be compactified by its ${\aleph_0}$-tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a subset of its ${\aleph_0}$-tangles. The ${\aleph_0}$-tangles of a graph are shown to form an inverse limit of the ultrafilters on the sets of components obtained by deleting a finite set of vertices. The ${\aleph_0}$-tangles that are ends are precisely the limits of principal ultrafilters. The ${\aleph_0}$-tangles that correspond to a highly connected part, or $\aleph_0$-block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations.