TITLE: When does graph minor containment imply subgraph containment?
QUESTION [15 upvotes]: Consider a path of length 3.  Any graph G which contains this graph as a minor must also contain it as a subgraph. For paths of any length this is easy to prove.
In general this happens for any graph in which each connected component is either a path or a subdivision of the claw grah. (The claw graph is the star graph on 4 vertices, or the complete bipartite graph $K_{1,3}$.) 
Does anyone know where I can find a proof of this fact? I know how to prove it, but if it has already appeared in some book/paper, it's easier for me to cite the result instead.
I am also interested in the following more general question, whose answer I do not know: How does one characterize sets of graphs H={H1,H2,...,Hk}, such that containing any of the Hi as a minor is equivalent to containing some graph from a finite set G={G1,G2,...,Gm} as a subgraph. In other words, for which sets H is H-minor containment equivalent to G-subgraph containment for some finite set G. (Note that this is trivial if we allow G to be infinite.)
For example, containing any graph from the set {path of length 3, claw} as a minor is equivalent to containing any graph from that set as a subgraph. As a non-trivial example, containing any from from {path of length 4, cycle of length 3} as a minor is equivalent to containing one of {path of length 4, cycle of length 3, cycle of length 4} as a subgraph.
Edit: For the single graph problem, I think I have stated a complete characterization of such graphs. I only wish to know if this appears in the literature somewhere. For the second problem I do not know a characterization (other than some special cases), and would welcome any information about the problem.

REPLY [4 votes]: The case when H has more than one forbidden minor is messy, but here's a partial analysis.
If H contains a linear forest (disjoint union of paths) then some path P_k is forbidden and the H-minor-free graphs are obviously describable by a finite set of forbidden subgraphs (P_k and the expansions of H that do not have a P_k subgraph).
If H does not contain a linear forest or disjoint union of subdivided claws, then the H-minor-free graphs include all subdivided claws. In this case the H-minor-free graphs are obviously not describable by a finite set of forbidden subgraphs -- any minimal forbidden minor of H can have its vertices blown up to degree-three trees and then its edges blown up to long paths so that the local neighborhood of any vertex looks the same as a subdivided claw, producing a graph that is not H-minor-free but looks locally like an H-minor-free graph.
If H does contain a cycle, and doesn't contain a linear forest, then the H-minor-free graphs are again obviously not describable by a finite set of forbidden subgraphs -- long cycles are not H-minor-free but look locally like paths, which are H-minor-free.
If H doesn't contain a linear forest or a cycle, but does contain a linear forest + claw in which only one of the claw edges is subdivided, then the H-minor-free graphs are describable by a finite set of forbidden subgraphs (the subdivided claw and the expansions of H that do not contain it -- because H doesn't contain a path or a cycle, each edge in each forbidden minor of H can only be expanded to a length shorter than the subdivided edge of the forbidden claw).
The remaining case is when H contains linear forest + claw subgraphs, but only those in which two or three of the claw edges have been subdivided, or when it contains graphs with more than one claw component.