Straight-line Drawings of 1-Planar Graphs

09/03/2021

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by   Franz J. Brandenburg, et al.

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A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which 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, so that edges of the same color do not cross. Hence, 1-planar graphs have geometric thickness two. In addition, each edge is crossed by edges with a common vertex if it is crossed more than twice. The drawings use high precision arithmetic with numbers with O(n log n) digits and can be computed in linear time from a 1-planar drawing