# Are there non trivial 2-basis of a 2-connected planar graph?

Based on MacLane's planarity criterion, planar graphs are exactly those that admit a 2-basis. Such basis can be easily obtained considering $$|E| - |V| + 1$$ of its faces. Denoting those basis as trivial bases, my question is: are there examples of non-trivial 2-basis of a 2-connected planar graph?

A non-trivial basis should contain at least one cycle surrounding at least two faces. I could neither find an example nor prove their inexistence.

Any help would be highly appreciated.

• Do you require that the degree of all vertices is at least 3? I think there are counterexamples if you allow degree 2 vertices. – Peter Shor Sep 17 '20 at 17:30
• Yes, the vertices may have degree 2 – Manuel Dubinsky Sep 18 '20 at 13:38

However, a planar graph that is 2-connected but not 3-connected can have multiple embeddings and when it does, the cycles of any given 2-basis may not be faces of a different embedding. As an example, $$K_{2,4}$$ has three combinatorially-distinct embeddings, for the three ways of choosing which pairs of degree-two vertices are non-co-facial, and for any particular embedding it is possible to find a cycle basis using one or both of the two non-facial cycles. (The degree-two vertices of this example can be replaced by two triangles connected edge-to-edge to make an example where all degrees are greater than two.)