A Tura´n-type theorem on chords of a convex polygon
Journal of Combinatorial Theory Series B
Graph Drawings with no k Pairwise Crossing Edges
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Topological Graphs with No Large Grids
Graphs and Combinatorics
On the maximum number of edges in topological graphs with no four pairwise crossing edges
Proceedings of the twenty-second annual symposium on Computational geometry
Crossing stars in topological graphs
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Extremal problems on triangle areas in two and three dimensions
Journal of Combinatorial Theory Series A
Drawing Graphs with Right Angle Crossings
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Notes on large angle crossing graphs
CATS '10 Proceedings of the Sixteenth Symposium on Computing: the Australasian Theory - Volume 109
Graphs that admit right angle crossing drawings
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
On the size of graphs that admit polyline drawings with few bends and crossing angles
GD'10 Proceedings of the 18th international conference on Graph drawing
Graphs that admit right angle crossing drawings
Computational Geometry: Theory and Applications
GD'11 Proceedings of the 19th international conference on Graph Drawing
Graphs That Admit Polyline Drawings with Few Crossing Angles
SIAM Journal on Discrete Mathematics
Counting plane graphs: cross-graph charging schemes
GD'12 Proceedings of the 20th international conference on Graph Drawing
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A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1-9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n-O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5n-O(1), thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.