Forbidden minors and subdivisions for toroidal graphs with no K 3,3 ’s Andrei Gagarin 2 Laboratoire de Combinatoire et d’Informatique Math´ematique (LaCIM) Universit´e du Qu´ebec ` a Montr´eal (UQAM) Montr´eal, Qu´ebec, Canada Wendy Myrvold 1,3 , John Chambers Department of Computer Science University of Victoria Victoria, British Columbia, Canada Abstract Forbidden minors and subdivisions for toroidal graphs are numerous. In contrast, the toroidal graphs with no K 3,3 ’s have a nice explicit structure and short lists of obstructions. For these graphs, we provide the complete lists of four forbidden minors and eleven forbidden subdivisions. Keywords: toroidal graph, embedding in a surface, forbidden minor, forbidden subdivision 1 Supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. 2 Email: gagarin@lacim.uqam.ca 3 Email: wendym@cs.uvic.ca