Consequences of cycle space cut space duality

Question

The cycle space cut space duality theorem for planar graphs states that:

The cycle space of a planar graph is the cut space of its dual graph, and vice versa.

I wish to know any consequences and applications of this result with possible proofs or references to them.

Answer 1

From Gomory-Hu trees:

In planar graphs, the Gomory–Hu tree is dual to the minimum weight cycle basis, in the sense that the cuts of the Gomory–Hu tree are dual to a collection of cycles in the dual graph that form a minimum-weight cycle basis.$^4$