Ends as tangles

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 $.