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There are uncountably many topological types of locally finite trees. (English) Zbl 1094.05019

Summary: Consider two locally finite rooted trees as equivalent if each of them is a topological minor of the other, with an embedding preserving the tree-order. Answering a question of van der Holst, we prove that there are uncountably many equivalence classes.

MSC:

05C05 Trees
05C10 Planar graphs; geometric and topological aspects of graph theory
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References:

[1] Diestel, R., Graph Theory (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1074.05001
[2] H. van der Holst, Problem posed at the 2005 Graph Theory workshop at Oberwolfach; H. van der Holst, Problem posed at the 2005 Graph Theory workshop at Oberwolfach
[3] Kühn, D., On well-quasi-ordering infinite trees—Nash-Williams’s theorem revisited, Math. Proc. Cambridge Philos. Soc., 130, 401-408 (2001) · Zbl 0985.05017
[4] Nash-Williams, C. St. J.A., On well-quasi-ordering infinite trees, Proc. Cambridge Philos. Soc., 61, 697-720 (1965) · Zbl 0144.23305
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