Sufficiency of sparsity conditions for remaining wallpaper groups

Determine whether the sparsity conditions consisting of overall (2,1)-tightness, (2,2)-sparsity for purely periodic subgraphs, and (2,3)-sparsity for balanced subgraphs are sufficient for generic forced symmetric rigidity for wallpaper groups pmg, pgg, p3m1, and p31m.

Background

The paper establishes a complete combinatorial characterization for the orientation-reversing wallpaper group pm and derives corresponding characterizations for its subgroups cm and pg. It explains that analogous sparsity conditions are insufficient for wallpaper groups containing even dihedral subgroups, namely pmm, cmm, p4m, p4g, and p6m, because of Bottema’s Mechanism gain graphs. The remaining groups pmg, pgg, p3m1, and p31m are identified as cases where the sufficiency of the same sparsity conditions is unresolved.

References

It is not known whether the sparsity conditions are sufficient for $pmg$, $pgg$, $p3m1$ or $p31m$.

— Orientation-Reversing Crystallographic Rigidity  (2502.14392 - Esson et al., 20 Feb 2025) in Section 5.1, “Other wallpaper groups”