Existence of a plane without edge crossings in projections of the random geometric graph
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910054360285184 |
|---|---|
| author | de Jonge, Lianne Nagy, Kinga |
| author_facet | de Jonge, Lianne Nagy, Kinga |
| contents | Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different planes leads to different drawings, and in particular, potentially more or fewer edge crossings. In this paper, we prove that if the connection radius is smaller than a given threshold, the probability that there exists a plane with zero crossings tends to one as $t\to \infty$. We also state the asymptotic probability that such a plane is found after considering a given number of randomly chosen planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of a plane without edge crossings in projections of the random geometric graph de Jonge, Lianne Nagy, Kinga Probability 60F05, 60D05, 68R10 Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different planes leads to different drawings, and in particular, potentially more or fewer edge crossings. In this paper, we prove that if the connection radius is smaller than a given threshold, the probability that there exists a plane with zero crossings tends to one as $t\to \infty$. We also state the asymptotic probability that such a plane is found after considering a given number of randomly chosen planes. |
| title | Existence of a plane without edge crossings in projections of the random geometric graph |
| topic | Probability 60F05, 60D05, 68R10 |
| url | https://arxiv.org/abs/2507.10389 |