Integer realizations of disk and segment graphs

Abstract: A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. It can be seen that every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the integer grid and equal (integer) radius; and every segment graph can be realized by segments whose endpoints lie on the integer grid. Here we show that there exist disk graphs on $n$ vertices such that in every realization by integer disks at least one coordinate or radius is $2^{2^{\Omega(n)}}$ and on the other hand every disk graph can be realized by disks with integer coordinates and radii that are at most $2^{2^{O(n)}}$; and we show the analogous results for unit disk graphs and segment graphs. For (unit) disk graphs this answers a question of Spinrad, and for segment graphs this improves over a previous result by Kratochv\'{\i}l and Matou{\v{s}}ek.