The obstructions for toroidal graphs with no ’s

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.