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 $k$.

Cited by in corpus (1)

• Trees of tangles in infinite separation systems