# 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?

• When you say 4-connected do you mean wrt edge cuts or vertex cuts? – daniello Oct 1 '20 at 3:07
• I mean vertex connectivity. – Rahul Jain Oct 3 '20 at 21:57

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.
• the 3-clique sum decomposition is perhaps what the OP is referring to. Wouldn’t we get a decomposition into a $K_4$ and a graph with minimum degree at $4$ the first time we try the decomposition. Further would this be a strong $3$ cut in the sense of Demaine et al? – SamiD Oct 16 '20 at 6:52