Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Convex drawings of 3-connected plane graphs
GD'04 Proceedings of the 12th international conference on Graph Drawing
Minimum-width grid drawings of plane graphs
Computational Geometry: Theory and Applications
Proving or disproving planar straight-line embeddability onto given rectangles
GD'09 Proceedings of the 17th international conference on Graph Drawing
Small grid drawings of planar graphs with balanced bipartition
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Universal line-sets for drawing planar 3-trees
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
Straight-line drawings of outerplanar graphs in O(dnlogn) area
Computational Geometry: Theory and Applications
Small grid drawings of planar graphs with balanced partition
Journal of Combinatorial Optimization
Universal point sets for planar three-trees
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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Despite a long research effort, finding the minimum area for straight-line grid drawings of planar graphs is still an elusive goal. A long-standing lower bound on the area requirement for straight-line drawings of plane graphs was established in 1984 by Dolev, Leighton, and Trickey, who exhibited a family of graphs, known as nested triangles graphs, for which (2n/3 - 1) × (2n/3 - 1) area is necessary. We show that nested triangles graphs can be drawn in 2n2/9 + O(n) area when the outer face is not given, improving a previous n2/3 area upper bound. Further, we show that n2/9 + Ω(n) area is necessary for any planar straight-line drawing of a nested triangles graph. Finally, we deepen our insight into the 4/9n2-4/3n+1 lower bound by Dolev, Leighton, and Trickey, which is conjectured to be tight, showing a family of plane graphs requiring more area.