Mac Lane's planarity criterion states that a graph is planar if and only if its cycle space has a basis such that each edge is contained in at most two cycles. We call a basis with this property a 2-basis. Finding a 2-basis for a planar graph is easy; just find an embedding of the graph in the plane and use the faces of the graph as the cycle basis.

I am interested in the more general algebraic version of this problem. Given a finite dimensional vector space $V$ over $\mathbb{Z}_2$ and a basis $B$, can I determine if $V$ has a 2-basis? That is, does there exist an algorithm that takes $B$ as input and produces a 2-basis if one exists?

I have not encountered any references to the idea of a 2-basis outside of planar graphs. Have these bases been studied in any other context? What is known about them?