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 believe I have a proof that the connected $k$-partitions of the vertices of $G$ are dual to the set of 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).