$k$-fold circuits and coning in rigidity matroids

TL;DR

This paper introduces the concept of $k$-fold circuits in rigidity matroids to derive new structural results, including conditions for $k$-fold circuit properties and an extension of Whiteley's coning lemma, aiding in understanding graph independence.

Contribution

It generalizes the notion of double circuits to $k$-fold circuits and applies this to analyze the structure of the generic $d$-dimensional rigidity matroid, providing new tools and results.

Findings

Analysis of 2-sums in rigidity matroids

Conditions for $k$-fold circuit properties in $ _d$

Extension of Whiteley's coning lemma

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