$k$-fold circuits and coning in rigidity matroids
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915463712210944 |
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| author | Hewetson, John Jackson, Bill Nixon, Anthony Smith, Ben |
| author_facet | Hewetson, John Jackson, Bill Nixon, Anthony Smith, Ben |
| contents | In 1980 Lovász introduced the concept of a double circuit in a matroid. The 2nd, 3rd and 4th authors recently generalised this notion to $k$-fold circuits (for any natural number $k$) and proved foundational results about these $k$-fold circuits. In this article we use $k$-fold circuits to derive new results on the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. These results include analysing 2-sums, showing sufficient conditions for the $k$-fold circuit property to hold for $k$-fold $\mathcal{R}_d$-circuits, and giving an extension of Whiteley's coning lemma. The last of these allows us to reduce the problem of determining if a graph $G$ with a vertex $v$ of sufficiently high degree is independent in $\mathcal{R}_d$ to that of verifying matroidal properties of $G-v$ in $\mathcal{R}_{d-1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $k$-fold circuits and coning in rigidity matroids Hewetson, John Jackson, Bill Nixon, Anthony Smith, Ben Combinatorics Metric Geometry 52C25 In 1980 Lovász introduced the concept of a double circuit in a matroid. The 2nd, 3rd and 4th authors recently generalised this notion to $k$-fold circuits (for any natural number $k$) and proved foundational results about these $k$-fold circuits. In this article we use $k$-fold circuits to derive new results on the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. These results include analysing 2-sums, showing sufficient conditions for the $k$-fold circuit property to hold for $k$-fold $\mathcal{R}_d$-circuits, and giving an extension of Whiteley's coning lemma. The last of these allows us to reduce the problem of determining if a graph $G$ with a vertex $v$ of sufficiently high degree is independent in $\mathcal{R}_d$ to that of verifying matroidal properties of $G-v$ in $\mathcal{R}_{d-1}$. |
| title | $k$-fold circuits and coning in rigidity matroids |
| topic | Combinatorics Metric Geometry 52C25 |
| url | https://arxiv.org/abs/2508.18838 |