NP-completeness of the $\ell_1$-embedding problem for simple graphs as sphere-of-influence graphs
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917167515041792 |
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| author | Jabuka, Stanislav |
| author_facet | Jabuka, Stanislav |
| contents | In graph theory an interesting question is whether for a fixed choice of $p\in [0,\infty]$, all simple graphs appear as sphere-of-influence graphs in some Euclidean space with respect to the $\ell_p$ metric. The answer is affirmative for $p=\infty$, negative for any $p\in (0,\infty)$, and unknown for $p=1$. The result of this work shows that for the case of $p=1$, this embeddability question is a (Promise) NP-Complete problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | NP-completeness of the $\ell_1$-embedding problem for simple graphs as sphere-of-influence graphs Jabuka, Stanislav Metric Geometry 05C12, 68R10, 68R12 In graph theory an interesting question is whether for a fixed choice of $p\in [0,\infty]$, all simple graphs appear as sphere-of-influence graphs in some Euclidean space with respect to the $\ell_p$ metric. The answer is affirmative for $p=\infty$, negative for any $p\in (0,\infty)$, and unknown for $p=1$. The result of this work shows that for the case of $p=1$, this embeddability question is a (Promise) NP-Complete problem. |
| title | NP-completeness of the $\ell_1$-embedding problem for simple graphs as sphere-of-influence graphs |
| topic | Metric Geometry 05C12, 68R10, 68R12 |
| url | https://arxiv.org/abs/2512.20740 |