On the density of maximal 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity. Maximal 1-planar graphs have at most 4n−8 edges. We show that there are sparse maximal 1-planar graphs with only $\frac{45}{17} n + \mathcal{O}(1)$ edges. With a fixed rotation system there are maximal 1-planar graphs with only $\frac{7}{3} n + \mathcal{O}(1)$ edges. This is sparser than maximal planar graphs. There cannot be maximal 1-planar graphs with less than $\frac{21}{10} n - \mathcal{O}(1)$ edges and less than $\frac{28}{13} n - \mathcal{O}(1)$ edges with a fixed rotation system. Furthermore, we prove that a maximal 1-planar rotation system of a graph uniquely determines its 1-planar embedding.