Recognizing a DOG is Hard but not when it is Thin and Unit

Abstract

We define the notion of disk-obedience for a set of disks in the plane and give results for disk-obedient graphs (DOGs), which are disk intersection graphs (DIGs) that admit a planar embedding with vertices inside the corresponding disks. We show that in general it is hard to recognize a DOG, but when the DIG is thin and unit (i.e., when the disks are unit disks), it can be done in linear time.