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 ω whose induced tangles of order k are closed. They are precisely the tangles τ for which there is a set of k vertices that decides τ by majority vote. Such a set exists if and only if the vertex degree plus the number of dominating vertices of ω is at least k.

Trees of tangles in infinite separation systems

“…This theorem is a cornerstone in Carmesin's proof that every infinite graph has a treedecomposition displaying all its topological ends. For more about the relation between ends and tangles also see [4,19]. We deduce Theorem 1•4 from Theorem 1•3 in Section 6•2•1.…”

Trees of tangles in infinite separation systems

“…This theorem is a cornerstone in Carmesin's proof that every infinite graph has a tree-decomposition displaying all its topological ends. For more about the relation between ends and tangles also see [11,18]. We deduce Theorem 1.4 from Theorem 1.3 in Section 6.2.1.…”