The obstructions for toroidal graphs with no K3,3K3,3’s

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3K3,3-subdivisions that coincide with the toroidal graphs with no K3,3K3,3-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 K3,3K3,3’s and prove that the lists are sufficient.