S. Hong, P. Eades, N. Katoh, G. Liotta, P. Schweitzer and Y. Suzuki, "A Linear-Time Algorithm for Testing Outer-1-Planarity", Algorithmica, DOI 10.1007/s00453-014-9890-8, Springer, May 2014.
A graph is 1-planar if it can be embedded in the plane with at most one
crossing per edge. It is known that the problem of testing 1-planarity of a graph is NP-complete.
In this paper, we study outer-1-planar graphs. A graph is outer-1-planar if
it has an embedding in which every vertex is on the outer face and each edge has at
most one crossing. We present a linear time algorithm to test whether a given graph
is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding
in linear time if it exists.