Theory of linear and integer programming
Theory of linear and integer programming
String graphs. II.: Recognizing string graphs is NP-hard
Journal of Combinatorial Theory Series B
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
Intersection graphs of segments
Journal of Combinatorial Theory Series B
Simple linear time recognition of unit interval graphs
Information Processing Letters
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
On Intersection Graphs of Segments with Prescribed Slopes
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
Algorithmic aspects of constrained unit disk graphs
Algorithmic aspects of constrained unit disk graphs
Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Convex polygon intersection graphs
GD'10 Proceedings of the 18th international conference on Graph drawing
Complexity of some geometric and topological problems
GD'09 Proceedings of the 17th international conference on Graph Drawing
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We give the first lower bounds on the grid size needed to represent the intersection graphs of~convex polygons. Here each corner of a polygon in the representation must lie on a corner of the grid. We provide a series of geometric constructions showing that for intersection graphs of: translated copies of any fixed parallelogram, grids of size Ω(n2) x Ω(n2) are needed; translated copies of any other fixed convex polygon, grids of size 2Ω(n) x 2Ω(n) are needed; homothetic copies of any fixed convex polygon, grids of size 2Ω(n) x 2Ω(n) are needed. We complement these results by giving a matching upper bound in each case. Hence we settle the complexity of the integer representation problem for these graphs. The upper bounds substantially improve earlier bounds and extend to higher dimensions.