Complexity classification of unit disk contact graph recognition

Determine whether recognizing unit disk contact graphs is ER-hard or ER-complete, and classify its computational complexity.

Background

A contact graph of unit disks has vertices representing unit disks and edges indicating tangency without overlap. While Koebe’s theorem characterizes disk contact graphs for planar graphs, restricting to unit-radius disks complicates recognition.

The survey notes that the problem is NP-hard but its ER-hardness remains unestablished.

References

However, if we require that disks have unit radius, the recognition problem is known to be NP-hard, but not known to be -hard~\ourref[Disk Graph!Unit Disk Contact Graph (Open)]{p:unitdiskcontact}.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Section 'Intersection Graphs'