Abstract
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.
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Brandenburg, F.J., Eppstein, D., Gleißner, A., Goodrich, M.T., Hanauer, K., Reislhuber, J. (2013). On the Density of Maximal 1-Planar Graphs. In: Didimo, W., Patrignani, M. (eds) Graph Drawing. GD 2012. Lecture Notes in Computer Science, vol 7704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36763-2_29
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