  Let G be a cubic graph, with girth at least five, such that for every
partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges
between X and Y. We prove that if there is no homeomorphic embedding of the
Petersen graph in G, and G is not one particular 20-vertex graph, then either
G\v is planar for some vertex v, or G can be drawn with crossings in the plane,
but with only two crossings, both on the infinite region. We also prove several
other theorems of the same kind.
