# Reference on generalization of plane graph duality between bonds and simple cycles

I've also asked this question on Mathoverflow, but it hasn't gotten an answer after several months: https://mathoverflow.net/questions/316132/reference-on-generalization-of-plane-graph-duality-between-bonds-and-simple-cycl

Let $$G$$ be a plane graph, and $$G^*$$ its dual. Among the $$k$$ partitions of the nodes of $$G$$, I'll call the connected k-partitions those such that each block of nodes of the partition induces a connected subgraph of $$G$$.

It is well known that the connected 2-partitions of the vertices $$G$$ are dual to the simple cycles of $$G^*$$. (The duality is by sending the partition to the cut edges of that partition. This is also called the bond / simple cycle duality.)

I can prove (it is not hard, unless I made a mistake) that the connected $$k$$-partitions of the vertices of $$G$$ are dual to the set of edge subgraphs $$K$$ of $$G^*$$ with

• $$H_1(K)$$ is of rank $$k - 1$$
• Each connected component $$K$$ is $$2$$ edge connected.

Again, the duality is by sending a partition to the cut edges in that partition. In the other direction, it comes by taking the connected components of $$G$$ after removing the edges crossing the subgraph $$K$$.

I use this result in the course of some other proof, and I'm looking for a reference for it (as this seems like the kind of thing that is either wrong, or well known).