Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time
  Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988].   In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be re-embedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges.   We give a linear-time algorithm for finding a straight-line drawable 1-plane re-embedding or the forbidden configuration. 
Seok-Hee Hong, Hiroshi Nagamochi
cs.CG cs.DS