Excluding a Long Double Path Minor

The “height” of a graphGis defined to be the number of steps to constructGby two simple graph operations. LetBnbe the graph obtained from ann-edge path by doubling each edge in parallel. Then, for any minor-closed class G of graphs, the following are proved to be equivalent: (1) SomeBnis not in G; (2) There is a numberhsuch that every graph in G has height at mosth; (3) G is well-quasi-ordered by the topological minor relation; (4) There is a polynomial functionp(·) such that the number of paths of every graphGin G is at mostp(|V(G)|+|E(G)|).