On plane rigidity matroids

TL;DR

This paper investigates properties of rigidity matroids in two dimensions, establishing uniqueness results, extending combinatorial characterizations, and classifying cubic graphs with specific orientation properties.

Contribution

It proves the uniqueness of the 2-rigidity matroid family without $K_{3,3}$ circuits, extends Bernstein's theorem to positive characteristic, and classifies cubic graphs based on rigidity independence.

Findings

$ ext{R}$ is the unique 2-rigidity matroid family without $K_{3,3}$ as a circuit.

Extended Bernstein's theorem to positive characteristic fields.

Classified all connected cubic graphs as having certain orientation and partition properties.

Abstract

We prove several results about matroids and matroidal families associated with rigidity in dimension $2$. In particular, we establish new properties of the generic rigidity matroid family $\mathcal{R}$ and Kalai's hyperconnectivity matroid family $\mathcal{H}$. We show that $\mathcal{R}$ is the unique matroidal $2$-rigidity family in which $K_{3,3}$ is not a circuit. As a geometric corollary of this result and the Bolker-Roth theorem, it follows that $\mathcal{H}$ and $\mathcal{R}$ are the only $2$-rigidity families associated with algebraic curves in $\mathbb{R}^2$. Bernstein used tropical geometry to characterize $\mathcal{H}$-independent graphs as those admitting an edge-ordering without directed cycles and alternating closed trails. We provide a combinatorial proof of the sufficiency direction and extend Bernstein's theorem to positive characteristic. It follows that the wedge power matroid of $n$ generic points in dimension $n-2$ does not depend on the field characteristic. Our proof method allows to identify many graphs that are independent in every $2$-rigidity family. In particular, we show this for all connected cubic graphs, with exceptions of $K_4$ and $K_{3,3}$. This gives a complete classification of cubic graphs in this respect and answers a question of Kalai in a strong form. As a corollary, we obtain a new property of cubic graphs: every connected cubic graph except $K_4$ and $K_{3,3}$ has an orientation without directed and alternating cycles. Equivalently, it can be edge-partitioned into two forests in a special `interlocked' way.