A Linear-Time Algorithm for Testing Outer-1-Planarity - Hong, Seok-Hee; Eades, Peter; Katoh, Naoki; Liotta, Giuseppe; Schweitzer, Pascal; Suzuki, Yusuke (Algorithmica, 2015)
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.