Structural properties of plane realization maxima

Determine whether every minimally rigid graph attaining the maximum complex realization count for a fixed number of vertices must satisfy any of the structural properties observed in the currently best-known plane examples.

Background

The authors list several properties shared by the currently best-known plane graphs, including degree constraints, the absence of adjacent degree-three vertices, nonplanarity for sufficiently large n, chromatic number three, and Hamiltonicity. They caution that the computational sample may be biased and explicitly leave unresolved whether such properties are necessary for genuinely maximal realization counts.

References

It is unclear so far on whether a graph with $2{G}=2{G}$ would need to have any of these properties indeed.

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 2, paragraph following the list of common properties of currently best-known plane graphs