Tangles are Decided by Weighted Vertex Sets

Abstract

We show that, given a $ k $-tangle $ \tau $ in a graph $ G $, there always exists a weight function $ w\colon V(G)\to\mathbb{N} $ such that a separation $ (A,B) $ of $ G $ of order $ {<}k $ lies in $ \tau $ if and only if $ w(A)<w(B) $, where $ w(U) := \sum_{u\in U}w(u) $ for $ U\subseteq V(G) $. We show that the same result holds also for tangles of hypergraphs as well as for edge-tangles of graphs, but not for edge-tangles of hypergraphs.