Complexity of penny graph recognition without triangles or induced 4-cycles

Determine whether recognizing penny graphs remains complete for the existential theory of the reals when restricted to graphs that exclude triangles and/or exclude induced 4-cycles; equivalently, ascertain if deciding whether a triangle-free and/or induced-4-cycle-free graph is a contact graph of unit disks in the plane is still ∃ℝ-hard.

Background

The paper’s ∃ℝ-hardness reduction for penny graph recognition relies on structural features such as 3-cycles (for rigidity) and induced 4-cycles (to simulate degree-3 vertices with variable angles). Hence it is unclear whether the same hardness persists when these substructures are forbidden. The authors note that penny graphs without 3-cycles are 2-degenerate (Eppstein), which could make encoding more challenging, and explicitly question whether hardness survives under these graph restrictions.