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Three well-known characterizations of planarity of graphs are the theorems of
Kuratowski, MacLane, and Whitney. The first is about forbidden subgraphs, the
second about a basis for the cycle space, and the third is about dual graphs.
Thomassen generalized Kuratowski's Theorem to "2-connected, compact, locally
connected metric spaces", Bruhn and Stein proved MacLane's Theorem for the
Freudenthal compactification of a locally finite graph. Bruhn and Diestel proved
a version of Whitney's Theorem for (compactifications of certain) infinite
graphs.
In this talk, we will see how to omit the "2-connected" hypothesis from
Thomassen's Theorem and generalize both of the other two theorems to compact
graph-like spaces.
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