Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
Discrete & Computational Geometry
Note: On the maximum number of edges in quasi-planar graphs
Journal of Combinatorial Theory Series A
Graphs that admit right angle crossing drawings
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
The straight-line RAC drawing problem is NP-hard
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
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We consider graphs that admit polyline drawings where all crossings occur at the same angle $\alpha\in (0,\frac{\pi}{2}]$. We prove that every graph on $n$ vertices that admits such a polyline drawing with at most two bends per edge has $O(n)$ edges. This result remains true when each crossing occurs at an angle from a small set of angles. We also provide several extensions that might be of independent interest.