The structure of K3, 3-subdivision-free toroidal graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K3,3-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K5-subdivision in [A. Gagarin and W. Kocay, “Embedding graphs containing K5-subdivisions”, Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K3,3-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, ”Counting labelled projective-planar graphs without a K3,3-subdivision”, submitted, arXiv:math.CO/ 0406140, (2004)].