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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.17861 |
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| _version_ | 1866913777504485376 |
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| author | Cote, Yazmin Uzcátegui-Aylwin, Carlos |
| author_facet | Cote, Yazmin Uzcátegui-Aylwin, Carlos |
| contents | This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on $\mathbb{Z}^2$, and the subsequent introduction of the topological digital plane $\mathbb{K}^2$ with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator $Γ^*$ transforming subsets of $\mathbb{Z}^2$ into subsets of $\mathbb{K}^2$. This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in $\mathbb{Z}^2$ correspond to topological properties in $\mathbb{K}^2$. Moreover, we address whether the topological Jordan curve theorem for $\mathbb{K}^2$ can be derived from the graph-theoretical version on $\mathbb{Z}^2$. Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17861 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bridging Graph-Theoretical and Topological Approaches: Connectivity and Jordan Curves in the Digital Plane Cote, Yazmin Uzcátegui-Aylwin, Carlos General Topology 54H30, 05C10, 68U03 This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on $\mathbb{Z}^2$, and the subsequent introduction of the topological digital plane $\mathbb{K}^2$ with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator $Γ^*$ transforming subsets of $\mathbb{Z}^2$ into subsets of $\mathbb{K}^2$. This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in $\mathbb{Z}^2$ correspond to topological properties in $\mathbb{K}^2$. Moreover, we address whether the topological Jordan curve theorem for $\mathbb{K}^2$ can be derived from the graph-theoretical version on $\mathbb{Z}^2$. Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology. |
| title | Bridging Graph-Theoretical and Topological Approaches: Connectivity and Jordan Curves in the Digital Plane |
| topic | General Topology 54H30, 05C10, 68U03 |
| url | https://arxiv.org/abs/2503.17861 |