Tangles are Decided by Weighted Vertex Sets

TL;DR

The paper addresses whether graph tangles can be decided by a weighted vertex set. It proves that for any finite graph $G=(V,E)$ and a $k$-tangle $\tau$, there exists a weight function $w:V(G)\to\mathbb{N}$ such that a separation $(A,B)$ of order $<k$ lies in $\tau$ if and only if $w(A)<w(B)$, with $w(U)=\sum_{u\in U}w(u)$. The proof uses a partial order on separations, analyzes maximal tangle separations via cross-counting, and applies Tucker's theorem to produce a suitable weight, with perturbations handling degenerate cases; the argument extends to tangles in hypergraphs and to regular $k$-profiles. It further extends to edge-tangles in graphs through a pairwise edge-weighting and a conversion to vertex weights, but a constructed hypergraph example shows that weighted deciders need not exist for all hypergraph edge-tangles, highlighting essential graph-hypergraph distinctions.

Abstract

We show that, given a $ k $-tangle $ τ$ 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 $ τ$ 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.

Theorems & Definitions (14)

• Theorem 2
• Lemma 3
• proof
• Lemma 4: Tucker's Theorem TuckerDuality
• proof : Proof of Theorem \ref{['thm:tangledecider']}
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 8
• proof