The Graph Minor Theorem tells us that any set of graphs that is closed under:
- Vertex Deletion
- Edge Deletion
- Edge Contraction
have a finite set of "forbidden minors", such that if a graph G contains at least one of those minors, you know that G is not within your set.
Is there an analogous forbidden subgraph result for a set of graphs that is closed under edge contraction, but not closed under edge or vertex deletion?