Wagner's theorem is a result in graph theory that provides a characterization of planar graphs. Specifically, it states that a graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph \( K_{5} \) (the complete graph on five vertices) or a subdivision of the complete bipartite graph \( K_{3,3} \) (the complete bipartite graph with three vertices in each part).
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