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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2505.00163 |
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| _version_ | 1866908344311087104 |
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| author | Czabarka, Éva Helm, Alec Székely, László |
| author_facet | Czabarka, Éva Helm, Alec Székely, László |
| contents | A tanglegram of size n is a graph formed from two rooted binary trees with n leaves each and a perfect matching between their leaf sets. Tanglegrams are used to model co-evolution in various settings. A tanglegram layout is a straight line drawing where the two trees are drawn as plane trees with their leaf-sets on two parallel lines, and only the edges of the matching may cross. The tangle crossing number of a tanglegram is the minimum crossing number among its layouts. It is known that tanglegrams have crossing number at least one precisely when they contain one of two size 4 subtanglegrams, which we refer to as cross-inducing subtanglegrams. We show here that a tanglegram with exactly one cross inducing subtanglegram must have tangle crossing number exactly one, and ask the question whether the tangle-crossing number of tanglegrams with exactly k cross-inducing subtanglegrams is bounded for every k. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tanglegrams with a Unique 1-Crossing-Critical Subtanglegram have Tangle Crossing Number 1 Czabarka, Éva Helm, Alec Székely, László Combinatorics 05C10, 05C05, 05C62, 92B10 A tanglegram of size n is a graph formed from two rooted binary trees with n leaves each and a perfect matching between their leaf sets. Tanglegrams are used to model co-evolution in various settings. A tanglegram layout is a straight line drawing where the two trees are drawn as plane trees with their leaf-sets on two parallel lines, and only the edges of the matching may cross. The tangle crossing number of a tanglegram is the minimum crossing number among its layouts. It is known that tanglegrams have crossing number at least one precisely when they contain one of two size 4 subtanglegrams, which we refer to as cross-inducing subtanglegrams. We show here that a tanglegram with exactly one cross inducing subtanglegram must have tangle crossing number exactly one, and ask the question whether the tangle-crossing number of tanglegrams with exactly k cross-inducing subtanglegrams is bounded for every k. |
| title | Tanglegrams with a Unique 1-Crossing-Critical Subtanglegram have Tangle Crossing Number 1 |
| topic | Combinatorics 05C10, 05C05, 05C62, 92B10 |
| url | https://arxiv.org/abs/2505.00163 |