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.