A finite simple graph is a planar graph if and only if it contains no graph subdivision of the complete graph or the complete bipartite graph as a subgraph; equivalently, neither is a topological minor. The easy direction follows by tracing each replacement graph path as one drawn edge. The converse asserts that every failure of planarity contains one of these two patterns.[ref]
Graph Theory by Reinhard Diestel (Planar Graphs preview, §4.4, Theorem 4.4.6, printed p. 111) dates Kuratowski's theorem to 1930 and the equivalent graph minor characterization to 1937. These dates are that textbook's historical attribution.
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Kuratowski's theorem is a fundamental result in topology, specifically in the area of planar graphs. It characterizes the planarity of a graph using the concepts of subgraphs and Kuratowski's two examples of non-planar graphs.