Classification of 2-rigidity families

Classify all 2-rigidity families and determine whether the generic rigidity family \mathcal{R}_2 and the hyperconnectivity family \mathcal{H}_2 are the only such families.

Background

A 2-rigidity family is a family of finite graphs whose labelled copies form the independence families of abstract 2-rigidity matroids on every vertex set. The two known examples are the generic rigidity family \mathcal{R}_2 and the hyperconnectivity family \mathcal{H}_2.

The paper proves only a representability obstruction: any other 2-rigidity family eventually yields non-linearly-representable matroids as the number of vertices grows. Thus, the conjecture remains open in the unrestricted, potentially nonrepresentable setting.

References

For $d=2$ the two known $d$-rigidity families are the generic rigidity family $\mathcal{R}_2$ and the hyperconnectivity family $\mathcal{H}_2$, and it was conjectured in that no further $2$-rigidity families exist.

— Linear matroid products: a synthetic approach  (2609.16747 - Tyomkyn, 15 Sep 2026) in Section 1, Introduction, paragraph preceding Theorem~\ref{thm:representable-families}