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Kuratowski and Wagner theorems characterize planar graphs in terms of forbidden homeomorphic subgraphs and forbidden minors, respectively. It turns out that both forbidden sets are the same: $\{K_5,K_{3,3}\}$.

Are there other examples where a forbidden set defines the same class of graphs in both ways?

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up vote 6 down vote accepted

One example that springs to mind: A (simple) graph is a forest iff it has no $K_3$ as a minor, and also iff it has no $K_3$ as a topological minor (homeomorphic subgraph).

More generally, if the forbidden graphs have maximum degree 3, there's no difference between minors and topological minors (see Proposition 1.7.4 in Diestel's textbook).

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Thank you! that prop. wasn't there in the 3rd edition. – Diego Nov 8 '11 at 19:23
It is Proposition 1.7.2 in the second edition. – András Salamon Aug 3 '14 at 13:42

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