Unified Hanani-Tutte theorem

Abstract

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.