On plane rigidity matroids

TL;DR

This work surveys foundational and new results on abstract $2$-rigidity matroids, focusing on the two central families $\mathcal{R}$ and $\mathcal{H}$ and their interactions with algebraic curves, combinatorial orientations, and graph structure. It provides a combinatorial proof of Bernstein’s sufficiency for $\mathcal{H}$-independence, extends it to positive characteristic, and proves that the wedge-power duality yields a characteristic-independent $W_n(n-2,p)$. A key outcome is the complete classification of cubic graphs with respect to $2$-rigidity independence, showing that all connected cubic graphs except $K_4$ and $K_{3,3}$ are independent in every $2$-rigidity family, and establishing the uniqueness of $\mathcal{R}$ as the only $2$-rigidity family avoiding $K_{3,3}$ as a circuit. The paper also situates these results in the broader context of matroidal families, poses open questions about the existence of further $2$-rigidity families, and discusses implications for orientations and decompositions of graphs, with potential impact on rigidity theory, tropical geometry, and graph algorithms.

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.

Theorems & Definitions (59)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Theorem 1.10: Ber17