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arXiv · 2508.18838

$k$-fold circuits and coning in rigidity matroids

John Hewetson·Bill Jackson·Anthony Nixon·Ben Smith
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Abstract

In 1980 Lov\'{a}sz 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 $\mathcal{R}_{d-1}$.

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BibTeXRIS

John Hewetson, Bill Jackson, Anthony Nixon, Ben Smith. 2025-08-26. $k$-fold circuits and coning in rigidity matroids. https://arxiv.org/abs/2508.18838

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