Classification of Minimal Separating Sets of Low Genus Surfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911058166284288 |
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| author | Aagaard, Christopher N. Veerman, J. J. P. |
| author_facet | Aagaard, Christopher N. Veerman, J. J. P. |
| contents | A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such sets show up in a variety of contexts. For example, in a wide class of metric spaces, if we choose distinct points p and q, then the set of points x satisfying d(x, p) = d(x, q) is a minimal separating set. In this paper we classify which topological graphs can be realized as minimal separating sets in surfaces of low genus. In general the question of whether a graph can be embedded at all in a surface is a difficult one, so our work is partly computational. We classify graphs embeddings which are minimal separating in a given genus and write a computer program to find all such embeddings and their underlying graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02357 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of Minimal Separating Sets of Low Genus Surfaces Aagaard, Christopher N. Veerman, J. J. P. Combinatorics A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such sets show up in a variety of contexts. For example, in a wide class of metric spaces, if we choose distinct points p and q, then the set of points x satisfying d(x, p) = d(x, q) is a minimal separating set. In this paper we classify which topological graphs can be realized as minimal separating sets in surfaces of low genus. In general the question of whether a graph can be embedded at all in a surface is a difficult one, so our work is partly computational. We classify graphs embeddings which are minimal separating in a given genus and write a computer program to find all such embeddings and their underlying graphs. |
| title | Classification of Minimal Separating Sets of Low Genus Surfaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.02357 |