Decomposing Single Crossing Minor Free graphs We know that for a single-crossing graph H of size h, there exists a constant ch whose value depends only on h, such that every four-connected component of an H-minor-free graph is either a planar graph or of treewidth at most ch.
I would like to know if some sort of the reverse of this also holds. That is, can we say for every such graph G whose four-connected components are of treewidth is at most w or a planar graph, there exists a constant cw whose value depends only on w and single-crossing graphs H1, H2, ..., Hl each of size at most cw, such that G belongs to the union of H_i-minor free-graphs for each i in 1, 2, ..., l?
If I understood your statement correctly, then it is false:
In a graph of maximum degree $3$ the $4$-connected components are all single vertices (and therefore planar AND of treewidth $\leq 1$). However we can make graphs of maximum degree 3 that contain arbitrarily large cliques as minors.