A Direct Proof of the Strong Hanani-Tutte Theorem on the Projective   Plane
  We reprove the strong Hanani-Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives hope to extend it to other surfaces. Moreover, our approach can be used to provide an efficient algorithm turning a Hanani-Tutte drawing on the projective plane into an embedding. 
\'Eric Colin de Verdi\`ere, Vojt\v{e}ch Kalu\v{z}a, Pavel Pat\'ak,   Zuzana Pat\'akov\'a and Martin Tancer
cs.CG math.CO