Straight-line drawings of 1-planar graphs

Abstract

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges such that edges of the same color do not cross. Thus 1-planar graphs have geometric thickness two. In addition, the drawing is nearly 1-planar, that is, it is 1-planar if all fan-crossed edges are removed. An edge is fan-crossed if it is crossed by edges with a common vertex if it is crossed more than twice. The drawing algorithm uses high precision arithmetic with numbers with O(nlog⁡n) digits and computes the straight-line drawing from a 1-planar drawing in linear time on a real RAM.