Theory of linear and integer programming
Theory of linear and integer programming
Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Uniform oriented matroids without the isotopy property
Discrete & Computational Geometry
Coordinate representation of order types requires exponential storage
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Discrete Mathematics - Topics on domination
String graphs requiring exponential representations
Journal of Combinatorial Theory Series B
Some provably hard crossing number problems
Discrete & Computational Geometry
Implicit representation of graphs
SIAM Journal on Discrete Mathematics
Representations of planar graphs
SIAM Journal on Discrete Mathematics
Intersection graphs of segments
Journal of Combinatorial Theory Series B
Unit disk graph recognition is NP-hard
Computational Geometry: Theory and Applications - Special issue on geometric representations of graphs
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Representing graphs by disks and balls (a survey of recognition-complexity results)
Discrete Mathematics
Plane integral drawings of planar graphs
Discrete Mathematics
Lectures on Discrete Geometry
JCDCG '98 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Algorithmic aspects of constrained unit disk graphs
Algorithmic aspects of constrained unit disk graphs
Robust algorithms for restricted domains
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
Journal of Combinatorial Theory Series B
Straight line embeddings of cubic planar graphs with integer edge lengths
Journal of Graph Theory
The number of bits needed to represent a unit disk graph
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Sphere and Dot Product Representations of Graphs
Discrete & Computational Geometry
A robust PTAS for maximum weight independent sets in unit disk graphs
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
A (4+ε)-approximation for the minimum-weight dominating set problem in unit disk graphs
WAOA'09 Proceedings of the 7th international conference on Approximation and Online Algorithms
Complexity of some geometric and topological problems
GD'09 Proceedings of the 17th international conference on Graph Drawing
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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. 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^^^@W^^^(^^^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 Kratochvil and Matousek.