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Complexity at maximum degree 3 and 4 for penny graph recognition

Determine the computational complexity of recognizing penny graphs restricted to graphs of maximum degree 3 and, separately, to graphs of maximum degree 4.

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Background

The authors show ∃ℝ-hardness of penny graph recognition and observe that with additional care the hardness can be maintained for maximum degree 5. Since every graph of maximum degree 2 is trivially a penny graph, the precise complexity for maximum degree 3 and 4 remains undetermined.

References

Since all graphs of maximum degree 2 are penny graphs, this leaves open the cases of maximum degree 3 and 4.

— 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)