Ends as tangles

TL;DR

This paper extends end-tangle theory from EndsAndTangles to characterize when end-induced tangles, restricted to finite-order separations $S_k$, are closed. It proves a sharp criterion: for an end $\omega$, the $k$-tangle $\tau_\omega \cap S_k$ is closed iff $\deg(\omega)+\operatorname{dom}(\omega)\ge k$, and relates this to finite relative deciders of size $k$. It introduces end cohesion $\mathrm{coh}(\omega)$ and classifies ends into infinite, unbounded, and bounded cohesion based on $\deg(\omega)$ and $\operatorname{dom}(\omega)$, yielding a precise description of when end tangles are topologically closed. The results connect global topological closure with local combinatorial witnesses (finite deciders) and highlight when infinite-cohesion ends admit an absolute decider, clarifying the interplay between end structure and tangles in infinite graphs.

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 $.

Theorems & Definitions (24)

• Example 3.1
• Example 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 1: EndsAndTangles
• proof
• Example 4.1
• Example 4.2