User blog comment:Hyp cos/SCG(n) and some related/@comment-1605058-20140814064902

Few notes/questions:


 * In your point about planar graphs, it sounds like plane embeddability is separate property of graph, while it's defining property of planarity. I'd suggest also simply saying that in planar embedding edges do not cross - set theoretic notation is far from necessary.


 * One can say more about CLGs - CLG is almost a dual graph, except it doesn't have vertex for outer face of the drawing.


 * It follows that CLG of CLG is exactly the staring graph with vertices of outer edge removed.


 * Suppose a planar graph G has two planar drawings, one with cycle level a, and other with cycle level b>a. Which category would we throw G into? To graphs with cycle level a or b? I suspect a would be the better choice.


 * Suppose that in embedding of G two cycles share two edges. Does its CLG have double edge where that happens? If so, second note about CLGs is incorrect.


 * "I don't know how SCG(n) goes on in outerplanar graphs" - correct me if I'm wrong, but I thought you showed that outerplanar graphs limit at BHO.


 * If you are ever going to go to the limit of SCG, I'm quite sure you'll require the notion of tree-width - if I'm not mistaken, induction on tree-width was key ingredient of proof of R-S theorem, so segregation according to tree-width would be meaningful.