Representing graphs by disks and balls (a survey of recognition-complexity results) Original Research Article

Practical applications, like radio frequency assignments, led to the definition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper, we survey recognition-complexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about bounded-ratio disk representations. We prove that the recognition of disk-intersection graphs (in the unbounded ratio case) is NP-hard. This result is proved in a more general setting of noncrossing arc-connected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unit-ball contact graphs is NP-hard in dimensions 3,4, and 8 (24).