$k$-fold circuits and coning in rigidity matroids

Abstract

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 ${\mathcal{R}}_{d-1}$.

27 pages, 7 figures. Minor improvements in clarity and exposition throughout