  Forbidden minors and subdivisions for toroidal graphs are numerous. We
consider the toroidal graphs with no $K_{3,3}$-subdivisions that coincide with
the toroidal graphs with no $K_{3,3}$-minors. These graphs admit a unique
decomposition into planar components and have short lists of obstructions. We
provide the complete lists of four forbidden minors and eleven forbidden
subdivisions for the toroidal graphs with no $K_{3,3}$'s and prove that the
lists are sufficient.

Combinatorics;Discrete Mathematics