A new proof of the 6 color theorem
Journal of Graph Theory
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
A Framework for Drawing Planar Graphs with Curves and Polylines
GD '98 Proceedings of the 6th International Symposium on Graph Drawing
Re-embeddings of Maximum 1-Planar Graphs
SIAM Journal on Discrete Mathematics
Right angle crossing graphs and 1-planarity
GD'11 Proceedings of the 19th international conference on Graph Drawing
Density of straight-line 1-planar graph drawings
Information Processing Letters
On sparse maximal 2-planar graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
Right angle crossing graphs and 1-planarity
Discrete Applied Mathematics
Parameterized complexity of 1-planarity
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
A linear time algorithm for testing maximal 1-planarity of graphs with a rotation system
Theoretical Computer Science
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A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity. Maximal 1-planar graphs have at most 4n−8 edges. We show that there are sparse maximal 1-planar graphs with only $\frac{45}{17} n + \mathcal{O}(1)$ edges. With a fixed rotation system there are maximal 1-planar graphs with only $\frac{7}{3} n + \mathcal{O}(1)$ edges. This is sparser than maximal planar graphs. There cannot be maximal 1-planar graphs with less than $\frac{21}{10} n - \mathcal{O}(1)$ edges and less than $\frac{28}{13} n - \mathcal{O}(1)$ edges with a fixed rotation system. Furthermore, we prove that a maximal 1-planar rotation system of a graph uniquely determines its 1-planar embedding.