Ends as tangles

TL;DR

This paper characterizes when tangles induced by ends of infinite graphs are closed, linking their properties to vertex sets that decide tangles by majority vote and relating this to vertex degrees and dominating vertices.

Contribution

It provides a complete characterization of ends with closed tangles of finite order in infinite graphs, connecting topological closure with combinatorial vertex properties.

Findings

Ends with closed tangles are exactly those decided by a vertex set of size k.

A set of k vertices decides a tangle if and only if the sum of vertex degree and dominating vertices is at least k.

The characterization applies to all finite k, linking topological and combinatorial properties.

Abstract

Every end of an infinite graph $ G $ defines a tangle of infinite order in $ G $. These tangles indicate a highly cohesive substructure in the graph if and only if they are closed in some natural topology. We characterize, for every finite $ k $, the ends $ \omega $ whose induced tangles of order $ k $ are closed. They are precisely the tangles $ \tau $ for which there is a set of $ k $ vertices that decides $ \tau $ by majority vote. Such a set exists if and only if the vertex degree plus the number of dominating vertices of $ \omega $ is at least $ k $.