Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912570233847808 |
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| author | Förster, Henry Kobourov, Stephen Miller, Jacob Zink, Johannes |
| author_facet | Förster, Henry Kobourov, Stephen Miller, Jacob Zink, Johannes |
| contents | A strengthened version of Harborth's well-known conjecture -- known as Kleber's conjecture -- states that every planar graph admits a planar straight-line drawing where every edge has integer length and each vertex is restricted to the integer grid. Positive results for Kleber's conjecture are known for planar 3-regular graphs, for planar graphs that have maximum degree 4, and for planar 3-trees. However, all but one of the existing results are existential and do not provide bounds on the required grid size. In this paper, we provide polynomial-time algorithms for computing crossing-free straight-line drawings of trees and cactus graphs with integer edge lengths and integer vertex position on polynomial-size integer grids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid Förster, Henry Kobourov, Stephen Miller, Jacob Zink, Johannes Computational Geometry Discrete Mathematics A strengthened version of Harborth's well-known conjecture -- known as Kleber's conjecture -- states that every planar graph admits a planar straight-line drawing where every edge has integer length and each vertex is restricted to the integer grid. Positive results for Kleber's conjecture are known for planar 3-regular graphs, for planar graphs that have maximum degree 4, and for planar 3-trees. However, all but one of the existing results are existential and do not provide bounds on the required grid size. In this paper, we provide polynomial-time algorithms for computing crossing-free straight-line drawings of trees and cactus graphs with integer edge lengths and integer vertex position on polynomial-size integer grids. |
| title | Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid |
| topic | Computational Geometry Discrete Mathematics |
| url | https://arxiv.org/abs/2509.04168 |