Abstract
For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of β΅~0~ disjoint copies of A is referred to as β΅~0~A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all __n__Ο΅β, but β΅~0~A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G~1~, G~2~,β¦ such that G~i~ is not a minor of G~j~ for all iββ βj. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all nβΟ΅ββ, then β΅~0~A is a minor of G, too. If true, this would be a farβreaching generalization of a classical result of R. Halin on families of disjoint oneβway infinite paths in graphs. Β© 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222β229, 2002; DOI 10.1002/jgt.10016