Kuratowski's Theorem states that a finite 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 the complete bipartite graph $K_{3,3}$ (three vertices on one side connected to three on the other). This theorem provides a fundamental characterization of planar graphs, linking their structure to certain forbidden subgraphs.
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