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Ends and tangles

Ends and tangles We show that an arbitrary infinite graph can be compactified by its $${\aleph _0}$$ ℵ 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 $${\aleph _0}$$ ℵ 0 -tangles. The $${\aleph _0}$$ ℵ 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 $${\aleph _0}$$ ℵ 0 -tangles that are ends are precisely the limits of principal ultrafilters.The $${\aleph _0}$$ ℵ 0 -tangles that correspond to a highly connected part, or $${\aleph _0}$$ ℵ 0 -block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Ends and tangles

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References (20)

Publisher
Springer Journals
Copyright
Copyright © 2017 by The Author(s)
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-016-0163-0
Publisher site
See Article on Publisher Site

Abstract

We show that an arbitrary infinite graph can be compactified by its $${\aleph _0}$$ ℵ 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 $${\aleph _0}$$ ℵ 0 -tangles. The $${\aleph _0}$$ ℵ 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 $${\aleph _0}$$ ℵ 0 -tangles that are ends are precisely the limits of principal ultrafilters.The $${\aleph _0}$$ ℵ 0 -tangles that correspond to a highly connected part, or $${\aleph _0}$$ ℵ 0 -block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Jan 18, 2017

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