Density of straight-line 1-planar graph drawings

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.