Drawing Graphs with Right Angle Crossings
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
2-Layer right angle crossing drawings
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Right angle crossing graphs and 1-planarity
GD'11 Proceedings of the 19th international conference on Graph Drawing
Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing
Journal of Graph Theory
On the density of maximal 1-planar graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
Testing maximal 1-planarity of graphs with a rotation system in linear time
GD'12 Proceedings of the 20th international conference on Graph Drawing
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A 1-planar drawing of a graph is such that each edge is crossed at most once. In 1997, Pach and Toth showed that any 1-planar drawing with n vertices has at most 4n-8 edges and that this bound is tight for n=12. We show that, in fact, 1-planar drawings with n vertices have at most 4n-9 edges, if we require that the edges are straight-line segments. We also prove that this bound is tight for infinitely many values of n=8. Furthermore, we investigate the density of 1-planar straight-line drawings with additional constraints on the vertex positions. We show that 1-planar drawings of bipartite graphs whose vertices lie on two distinct horizontal layers have at most 1.5n-2 edges, and we prove that 1-planar drawings such that all vertices lie on a circumference have at most 2.5n-4 edges; both these bounds are also tight.