Additionally they show that the ply of a planar cover may be arbitrarily large.
Negami (1986) proved, conversely, that if a connected graph H has a two-ply planar cover then H must have an embedding into the projective plane.
Negami (1988) proved that every connected graph with a planar regular cover can be embedded into the projective plane.
[12] Based on these two results, he conjectured that in fact every connected graph with a planar cover is projective.
[14] It is also known as Negami's "1-2-∞ conjecture", because it can be reformulated as stating that the minimum ply of a planar cover, if it exists, must be either 1 or 2.
[16] Since Negami made his conjecture, it has been proven that 31 of these 32 forbidden minors either do not have planar covers, or can be reduced by Y-Δ transforms to a simpler graph on this list.
[21] The conjecture is true for regular emulators, coming from automorphisms of the underlying covering graph, by a result analogous to the result of Negami (1988) for regular planar covers.
[22] However, several of the 32 connected forbidden minors for projective-planar graphs turn out to have planar emulators.
Finding a full set of forbidden minors for the existence of planar emulators remains an open problem.