The $d$-dimensional realisation number of a rigid graph
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912923182432256 |
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| author | Dewar, Sean Nixon, Anthony Smith, Ben |
| author_facet | Dewar, Sean Nixon, Anthony Smith, Ben |
| contents | Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary dimensions: (i) we prove that subgraph inclusion translates to realisation number divisibility; and (ii) we provide lower bounds on realisation numbers under specific graph operations in all dimensions. We use these methods to prove that every triangulated sphere with $n$ vertices has at least $2^{n-4}$ edge-length equivalent realisations in 3-dimensions, extending a 2-dimensional result of Jackson and Owen in the case of planar graphs. Additionally, our tools solve a family of conjectures set by Grasegger regarding how 1-extensions, X-replacements, and V-replacements affect realisation numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20766 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $d$-dimensional realisation number of a rigid graph Dewar, Sean Nixon, Anthony Smith, Ben Combinatorics Algebraic Geometry Metric Geometry 52C25, 05C10, 68R12, 14C17 Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary dimensions: (i) we prove that subgraph inclusion translates to realisation number divisibility; and (ii) we provide lower bounds on realisation numbers under specific graph operations in all dimensions. We use these methods to prove that every triangulated sphere with $n$ vertices has at least $2^{n-4}$ edge-length equivalent realisations in 3-dimensions, extending a 2-dimensional result of Jackson and Owen in the case of planar graphs. Additionally, our tools solve a family of conjectures set by Grasegger regarding how 1-extensions, X-replacements, and V-replacements affect realisation numbers. |
| title | The $d$-dimensional realisation number of a rigid graph |
| topic | Combinatorics Algebraic Geometry Metric Geometry 52C25, 05C10, 68R12, 14C17 |
| url | https://arxiv.org/abs/2602.20766 |