Excluding a Long Double Path Minor

The ``height'' of a graph G is defined to be the number of steps to construct G by two simple graph operations. Let Bn be the graph obtained from an n-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) Some Bn is not in G; (2) There is a number h such that every graph in G has height at most h; (3) G is well-quasiordered by the topological minor relation; (4) There is a polynomial function p( } ) such that the number of paths of every graph G in G is at most p( |V(G)|+|E(G)|).