Assertion: graph | /r/MadeOf | neither k
Facets:
value: as a subdivision
type: OTHER
value: by wagner's theorem
type: OTHER
Source sentences:
text: However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite graph K 3,3 as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions.