On 1-Planar Graphs with Rotation Systems

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. This causes essential distinctions to planar graphs: planarity can be tested in linear time whereas 1-planarity is NP-hard [11]. We improve this result and show the NP-hardness for 1-planar graphs with a given rotation system. In addition, the crossing number problem remains NP-hard for 1-planar graphs even with a rotation system. However, there are tractable cases: 1-planarity can be tested efficiently for embedded graphs and for maximal graphs with a given rotation system.

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