Jackson–Owen lower-bound conjecture for planar rigidity realisation numbers

Prove that every minimally 2-rigid graph with n vertices has 2-realisation number at least 2^{n-3}.

Background

The paper studies the 2-realisation number c_2(G), which is the generic number of complex planar realisations of a 2-rigid graph, counted up to congruence. The conjectured lower bound asserts exponential growth in the number of vertices for every minimally 2-rigid graph.

The paper notes that Jackson and Owen proved the conjecture for planar minimally 2-rigid graphs, and that it had been computationally verified for graphs with at most 12 vertices. However, outside those cases, the authors state that the only general lower bound currently available is c_2(G) greater than or equal to 2. Their realisation-basis construction supplies a further combinatorial lower bound, but the K_{3,3} example shows that this bound need not reach the conjectured value.

References

Jackson and Owen conjectured the following lower bound for the realisation number. Every minimally 2-rigid graph $G$ with $n$ vertices satisfies $c_2(G) \geq 2{n-3}$.

— A tropical approach to rigidity: counting realisations of frameworks  (2502.10255 - Clarke et al., 14 Feb 2025) in Section “Realisation bases,” immediately before Conjecture 1 (labelled conjecture:lowerbound)