Extension of the construction to hyperconnectivity matroids

Determine whether the main construction remains valid when the rigidity matroid family R_d is replaced by the hyperconnectivity matroid family H_d, including whether the constructed odd-order graphs are H_{2,n}-acceptable.

Background

The paper suggests that its construction may extend from rigidity matroids to hyperconnectivity matroids. Although coning preserves hyperconnectivity independence and circuits, the even-order graphs are not suitable bases and the circuit structure of the 2-hyperconnectivity matroid lacks a simple characterization. Consequently, it is unresolved whether the odd-order graphs satisfy the required acceptability conditions.

References

Secondly, although the odd-order constructed graphs given in \Cref{subsec:odd} are bases of 2-hyperconnectivity matroid (\Cref{prop:hypconn}), it is unclear if said graphs are $\mathcal{H}_{2,n}$-acceptable since the circuits of 2-hyperconnectivity matroid have no simple characterisation.

— Constructing pathological symmetric abstract rigidity matroids  (2610.12036 - Dewar et al., 8 Oct 2026) in Section 4, “Open problems,” subsection “Extending main results to hyperconnectivity and cofactor matroids”