# Characterizing graph classes by the same forbidden set in two ways

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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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