Integer realizations of disk and segment graphs

TL;DR

This paper investigates the complexity of integer realizations of disk, unit disk, and segment graphs, showing exponential lower bounds and upper bounds on the size of integer coordinates and radii needed for such realizations.

Contribution

It establishes exponential lower bounds and matching upper bounds for integer realizations of disk, unit disk, and segment graphs, answering open questions and improving previous results.

Findings

Existence of disk graphs requiring doubly exponential coordinates or radii in any integer realization.

All disk, unit disk, and segment graphs can be realized with doubly exponential size integer coordinates and radii.

Answers to open questions by Spinrad and improvements over prior work by Kratochvíl and Matoušek.

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.