User blog comment:Hyp cos/SCG(n) and some related/@comment-1605058-20140814064902/@comment-1605058-20140814163953

If your answer to bullet 5 is as you gave it, then forget about bullets 2 and 3, because it no longer holds.

You are wrong about outerplanar graphs - what you are thinking about are non-planar graphs. Outerplanar graph is a graph G such that, if we add additional vertex and connect it to every edge of G, then we will get a planar graph. Equivalently, there is a planar drawing of G with all its vertices on the outer face. Equivalently, G doesn't have K_4 or K_2,3 (the graph you have at BHO level) as a minor.

Treewidth is a number given to every graph, which expresses how much the graph is "tree-like". It's definition is quite technical, and I can't explain it too well, so I would recommend searching it online.