A linear time algorithm for testing maximal 1-planarity of graphs with a rotation system

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. 1-planarity is known NP-hard, even for graphs of bounded bandwidth, pathwidth, or treewidth, and for near-planar graphs in which an edge is added to a planar graph. On the other hand, there is a linear time 1-planarity testing algorithm for maximal 1-planar graphs with a given rotation system. In this work, we show that 1-planarity remains NP-hard even for 3-connected graphs with (or without) a rotation system. Moreover, the crossing number problem remainsNP-hard for 3-connected 1-planar graphs with (or without) a rotation system. Submitted: August 2014 Reviewed: January 2015 Revised: January 2015 Accepted: January 2015 Final: January 2015 Published: January 2015 Article type: Regular paper Communicated by: G. Liotta This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) grant Br835/18-1. E-mail addresses: [email protected] (Christopher Auer) [email protected] (Franz J. Brandenburg) [email protected] (Andreas Gleißner) [email protected] (Josef Reislhuber) 68 Auer et al. 1-Planarity of Graphs with a Rotation System