Three Theorems on Negami's Planar Cover Conjecture

A long-standing Conjecture of S.

Negami states that a connected graph has a finite planar cover if and only if it embeds in the projective plane.

It is known that the Conjecture is equivalent to the fact that \emph{the graph $K_{1,2, 2, 2}$ has no finite planar cover}.

We prove three theorems showing that the graph $K_{1,2, 2, 2}$ admits no planar cover with certain structural properties, and that the minimal planar cover of $K_{1,2, 2, 2}$ (if it exists) must be $4$-connected.