On plane rigidity matroids

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.

💡 Research Summary

This paper investigates two central matroid families that arise in two‑dimensional rigidity theory: the generic plane rigidity matroid 𝑹 and Kalai’s hyperconnectivity matroid 𝑯. Both families are “matroidal 2‑rigidity families,” meaning that for each number of vertices n they are symmetric graph matroids of rank 2n − 3, and the property of independence is hereditary across larger vertex sets. The authors first establish a striking uniqueness result: among all 2‑rigidity families, 𝑹 is the only one that does not contain the complete bipartite graph K₃,₃ as a circuit. Consequently, any 2‑rigidity family other than 𝑹 must treat K₃,₃ as a circuit. By invoking the Bolker–Roth theorem, they deduce that for any irreducible algebraic plane curve C, the associated rigidity matroid is 𝑯 when C is a conic (degree 2) and 𝑹 otherwise (degree ≥ 3). This gives a complete classification of algebraic‑curve‑induced 2‑rigidity matroids.

The second major theme concerns a combinatorial characterization of 𝑯‑independent graphs originally proved by Bernstein using tropical geometry. A graph is 𝑯‑independent if and only if it admits an orientation with no directed cycles and no alternating closed trails (a trail where consecutive edges share the same direction at each intermediate vertex). The authors provide a purely combinatorial proof of the sufficiency direction and extend the result to all characteristics p ≥ 0. They show that the hyperconnectivity matroids 𝑯ₙ(p) defined over a field of characteristic p coincide with 𝑯ₙ (characteristic 0). As a corollary, the wedge‑power matroid Wₙ(n − 2, p), which is the linear matroid of all 2‑vectors vᵢ∧vⱼ from n generic points in an (n − 2)‑dimensional space, is independent of the field characteristic.

The paper then turns to the behavior of connected cubic graphs (3‑regular graphs). Using the uniqueness of 𝑹 and the combinatorial tools developed for 𝑯, the authors prove that every connected cubic graph other than K₄ and K₃,₃ is independent in every 2‑rigidity family. Since K₄ is a circuit by definition and K₃,₃ is a circuit in all families except 𝑹, this yields a complete classification of cubic graphs with respect to 2‑rigidity families. An immediate graph‑theoretic corollary is that every such cubic graph admits an orientation with no directed cycles and no alternating cycles; equivalently, its edge set can be partitioned into two forests in a special “interlocked” way (each forest is a directed pseudoforest with maximum out‑degree 1). This answers a question posed by Kalai in a strong form.

Methodologically, the paper proceeds as follows. Section 2 gives the combinatorial proof that any Bernstein‑orientable graph is “uniquely forest‑partitionable” (UFP), which implies 𝑯‑independence in all characteristics. The proof constructs an auxiliary bipartite forest from a given acyclic orientation and uses parity arguments to derive a red/blue edge coloring with out‑degree ≤ 1 in each color class. Section 3 collects general properties of matroidal families, establishing hereditary and symmetry conditions. Sections 4 and 5 contain the proofs of the uniqueness of 𝑹 (Theorem 1.6) and the universal independence of cubic graphs (Theorem 1.8), respectively. The final section discusses implications, including the characteristic‑free nature of the wedge‑power matroid and potential extensions to higher dimensions or other regular graphs.

In summary, the authors achieve three major contributions: (1) a uniqueness theorem for the generic plane rigidity matroid, (2) a characteristic‑free combinatorial characterization of hyperconnectivity matroids, and (3) a complete classification of cubic graphs with respect to all 2‑rigidity matroids, together revealing new structural links between rigidity theory, algebraic geometry, tropical geometry, and classical graph theory.