Kuratowski's theorem given the necessary and sufficient conditions for it to hold for a graph and every graph with this property is four-colorable by the four-color theorem.
The first polynomial-time test for this property was written by Shih and Hsu in 1999, but it is still outperformed by the 1974 algorithm of Hopcroft and Tarjans in many cases. A famous instructive example regarding this, the Petersen star, does not possess it because removing an outer vertex creates an object homeomorphic to K-three-three. Kuratowski's theorem given the necessary and sufficient conditions for it to hold for a graph and every graph with this property is four-colorable by the four-color theorem. FTP, name this property of graphs that may be unambiguously embedded on a surface, or drawn without crossing a line or lifting the drawing implement.
The first polynomial-time test for this property was written by Shih and Hsu in 1999, but it is still outperformed by the 1974 algorithm of Hopcroft and Tarjans in many cases.
planarity