Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NP-hard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NP-hard. We conjecture that K-sphericity is NP-hard for all xed K greater than 1.
In REF , Breu and Kirkpatrick showed that the unit disk graph recognition problem is NP-hard.
Unit Disk Graph Recognition is NP-Hard
Computational Geometry. Theory and Applications
Mathematics Computer Science