Integer realizations of disk and segment graphs

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.

💡 Research Summary

The paper investigates the size of the integer grid required to realize three fundamental classes of intersection graphs in the plane: disk graphs, unit‑disk graphs, and segment graphs. A disk graph is defined as the intersection graph of (open) disks, a unit‑disk graph as the intersection graph of equal‑radius disks, and a segment graph as the intersection graph of line segments. It is well‑known that any graph from each class can be realized using disks (or segments) whose defining parameters are integers: disk centers can be placed on the integer lattice ℤ² and radii can be taken integer, unit‑disk graphs can be realized with a common integer radius, and segment graphs can be realized with endpoints in ℤ². The natural question, raised by Spinrad and later formalized as the Polynomial Representation Hypothesis (PRH) by van Leeuwen and van Leeuwen, asks whether the required integers can always be bounded by a polynomial (or more precisely by 2^{O(n^K)} for some constant K) in the number of vertices n.

The authors answer this question in the negative. They introduce three functions:

• f_DG(n) = max_{G∈DG(n)} min_{integer realizations of G} k(A), where k(A) is the smallest integer k such that the whole realization lies inside the square