Graphs That Admit Polyline Drawings with Few Crossing Angles

We consider graphs that admit polyline drawings where all crossings occur at the same angle α ∈ (0, π2 ]. 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.