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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 script G of graphs, the following are proved to be equivalent: (1) Some Bn is not in script G; (2) There is a number h such that every graph in script G has height at most h; (3) script G is well-quasi-ordered by the topological minor relation; (4) There is a polynomial function p( • ) such that the number of paths of every graph G in script G is at most p(\V(G)\ + \E(G)\). © 1996 Academic Press, Inc.

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Journal of Combinatorial Theory. Series B

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