Bar-and-joint rigidity on the moment curve coincides with cofactor rigidity on a conic

TL;DR

This paper establishes a deep connection between bar-and-joint rigidity and cofactor rigidity for points on the moment curve and conics, revealing their equivalence in these geometric settings and exploring differences in bipartite cases.

Contribution

It proves the equivalence of the bar-and-joint rigidity matroid and the hyperconnectivity matroid for points on the moment curve and conics, and analyzes their relationship in bipartite graphs.

Findings

Rigidity matroids coincide on the moment curve and conics.

Hyperconnectivity in dimension two is equivalent to moment curve points.

Bipartite graph rigidity matroids are more flexible than hyperconnectivity.

Abstract

We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the $C^{d-2}_{d-1}$-cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality. We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.