Necessary structural properties of extremal planar graphs

Determine whether minimally rigid planar graphs attaining the maximum number of complex realizations must satisfy any of the listed common properties, including minimum degree 3 and maximum degree 4, nonadjacency of degree-3 vertices, absence of 3-cycles for n greater than or equal to 12, nonplanarity, chromatic number 3, Hamiltonicity, and the requirement of a type E1c 1-extension at each degree-3 vertex.

Background

The currently best-known high-realization graphs share several structural features, but the computations were performed on selected data sets rather than on all minimally rigid graphs at larger sizes. The authors explicitly caution that these observed features may be sampling artifacts. The unresolved question is whether any of these properties are genuinely necessary for attaining the extremal 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”