In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who published it in 1937.
As a base case, G is a tree, then it has no bounded faces and C(G) is zero-dimensional and has an empty basis.
As a linearly independent set of the same size as the dimension of the space, this collection of cycles must form a basis.
If this is the case, then leaving any one of the cycles out produces a basis satisfying Mac Lane's formulation of the criterion.
If a planar graph is embedded on a sphere, its face cycles clearly satisfy Lefschetz's property.
This property is used in defining the meshedness coefficient of the graph, a normalized variant of the number of bounded face cycles that is computed by dividing the circuit rank by 2n − 5, the maximum possible number of bounded faces of a planar graph with the same vertex set (Buhl et al. 2004).