Decomposing graphs of genus one Planar graphs are $K_{3,3}$-free. Such graphs can be decomposed into tri-connected components, which are known to be either planar or $K_5$ components.


  Is there such a "nice" decomposition of graphs of genus one ?


In their seminal work on graph minors, Roberston and Seymour showed that every minor-free graph can be decomposed into a "clique-sum" of "almost planar" graphs. This, of course, applies to bounded-genus graphs also. I am looking for decompositions specific to graphs of genus one, to better understand their structural properties.
I think that Robertson and Seymour showed that every minor-free graph can be decomposed into a "clique-sum" of "almost bounded genus" graphs. The basic building blocks are not planar graphs but graphs of bounded genus (genus depending on the excluded minor). I think that toroidal graphs are not decomposable any further.