Vocabulary - Partition

The two groups in a bipartite graph are called partitions. You often label them $U$ and $V$. Every node belongs to exactly one partition. This partition is not unique. In a graph with two disconnected nodes, you could put both in $U$, or both in $V$, or one in each.

All are valid. What matters is that every edge crosses between partitions. For connected bipartite graphs, once you fix one node's partition, all others are determined. The structure forces every other node into a specific side.