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.
Decomposing graphs of genus one
graph theory;co.combinatorics;planar graphs;graph minor