Another proof of the result on rotation compatible planar covers

Negami's Planar Cover Conjecture asserts that a connected graph has a finite planar cover if and only if it can be embedded on the projective plane.

While this statement has already been proven for rotation compatible planar covers, namely covers equipped with a certain condition on the rotation system, the existing proof relies on advanced algebraic and topological methods.

In this paper, we provide another proof of this result, focusing primarily on combinatorial arguments based on a structural analysis with respect to a spanning tree in the base graph.