On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs. Bekos, M. A, Cornelsen, S., Grilli, L., Hong, S., & Kaufmann, M. ALGORITHMICA, 79:401–427, 2017.
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs [pdf]Paper  doi  abstract   bibtex   
Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a linear-time algorithm to test whether a given graph is maximal outer-fan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-complete, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given.

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