Tangles are Decided by Weighted Vertex Sets
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2018
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| _version_ | 1866912376923619328 |
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| author | Elbracht, Christian Kneip, Jay Lilian Teegen, Maximilian |
| author_facet | Elbracht, Christian Kneip, Jay Lilian Teegen, Maximilian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_06821 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Tangles are Decided by Weighted Vertex Sets Elbracht, Christian Kneip, Jay Lilian Teegen, Maximilian Combinatorics 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. |
| title | Tangles are Decided by Weighted Vertex Sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1811.06821 |