On the Representation of Disk Graphs

Abstract. For an arbitrary graph G, we consider the problem of deciding whether G is a disk graph (DG). The problem is known to be NP-hard, but it is open whether the problem actually is in NP. The problem is related to another open problem, the Polynomial Representation Hypothesis (PRH) for disk graphs: given an n-node disk graph G, can it be embedded in the plane such that the disk centers and disk radii have a binary representation in polynomially many,bits. We give several reductions of the problem, and prove that the PRH for disk graphs is equivalent to another interesting and purely geometric conjecture, the Separation Hypothesis for DGs. We give an exact, exponential algorithm for recognizing DG’s that have an -separated embedding, for any given > 0. Most results apply ipso facto to unit disk graphs, and will generalize to ball graphs and unit ball graphs of fixed sphericity d > 2.