Unified Hanani–Tutte Theorem

  • Radoslav Fulek
  • Jan Kynčl
  • Dömötör Pálvölgyi
Keywords: Hanani–Tutte theorem, Planar graph, Rotation system


We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in $D$. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.

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