Whitney's planarity criterion
Characterization of planar graphs by matroids / From Wikipedia, the free encyclopedia
In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney.[1] It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid).
In purely graph-theoretic terms, this criterion can be stated as follows: There must be another (dual) graph G′ = (V′,E′) and a bijective correspondence between the edges E′ and the edges E of the original graph G, such that a subset T of E forms a spanning tree of G if and only if the edges corresponding to the complementary subset E − T form a spanning tree of G′.