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arXiv · 1510.04050

Ends and Tangles

Reinhard Diestel
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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.

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Reinhard Diestel. 2015-10-14. Ends and Tangles. https://doi.org/10.1007/s12188-016-0163-0

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