I need a clarification on the question: a bipartite graph has a unique bipartition (up to changing two sets) iff it is connected
Specifically, I am not sure what it means by "unique bipartition".
Consider the following graph
The graph is disconnected since there isn't a path connecting the first bottom vertex to the second top vertex, but why isn't the bipartition (shown by the encircled)
unique?
Graph created using: https://www.draw.io/
Note: Ok I figured out the answer by the time I wrote this question, but feel free to provide answer to the benefit of other readers.
A bipartition can be formed by doing a proper 2-colouring of the vertices. If the graph is connected, up to colour choice there is only one such colouring. If the graph is disconnected with $n$ components, each component can have one of two colourings, leading to $2^{n-1}$ bipartitions up to swapping the two parts.
In the given example, a different bipartition can be formed by letting one part have the northwest and southeast vertices, the other taking the northeast and southwest vertices.