Characterization of maximum-realization graphs in the plane

Characterize whether every minimally rigid graph attaining the maximum number of complex realizations for a fixed vertex count must satisfy any of the structural properties observed in the currently best-known planar examples.

Background

The paper lists several common properties of currently best-known planar graphs, including degree constraints, nonadjacency of degree-three vertices, nonplanarity for sufficiently large n, chromatic number three, and Hamiltonicity. The authors caution that the computational sample may be biased and explicitly leave unresolved whether any of these properties are necessary for a graph achieving the maximum realization count.

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, subsection “Realizations in the plane”