M. A. Bekos, S. Cornelsen, L. Grilli, S. Hong and M. Kaufmann, "On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs", Proceedings of Graph Drawing 2014, pp. 198-209, LNCS, Springer, 2014.
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 polynomial-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-hard, for the case
where the rotation system (i.e., the cyclic order of the edges around each vertex) is given.