On vertex sets inducing tangles

TL;DR

This paper investigates whether every k-tangle in a graph can be represented by a vertex set or weight function, reducing the problem to bounded-size graphs and using topological minors to understand their structure.

Contribution

It reduces the question of vertex set representation of k-tangles to bounded-size graphs and establishes bounds on the size of vertex sets or weight functions inducing tangles.

Findings

Every k-tangle is induced by a weight function with bounded total weight.

The problem reduces to graphs of size bounded by a function in k.

Any k-tangle can be lifted from a topological minor of bounded size.

Abstract

Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$.