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Complexity of approximate penny graph recognition for constant radii ratio

Determine the computational complexity of recognizing contact graphs of disks whose radii are constrained to lie in a fixed constant interval [1, ρ] with ρ > 1 independent of the input size.

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Background

Breu and Kirkpatrick introduced the approximate penny graph recognition problem, proving NP-hardness when radii may vary over an interval. Using standard methods, the authors note that ∃ℝ-hardness holds when ρ approaches 1 with dependence on n (e.g., ρ = 2{-2{nc}}). However, the precise complexity when ρ is a fixed constant remains unresolved.

References

Using standard methods—see the case of approximate RAC-drawings, for example—it can be shown that the problem remains -hard for ρ = 2{-2{nc}} for some constant c>0, for n-vertex graphs, but the complexity of the approximate penny graph recognition problem in which ρ itself is constant is open.

— Recognizing Penny and Marble Graphs is Hard for Existential Theory of the Reals (2508.10136 - Lubiw et al., 13 Aug 2025) in Section 7 (Open Questions)