I'm looking for a specific name for a bipartite graph $(U,V,E)$ in which there is at most one edge incident to each vertex $u \in U$. That is, $|E_u| \le 1$ for all $u \in U$, where $E_u = \{(u,v) \in E\}$.
The best I have been able to think of is "star forest", but this term seems to be used specifically for subgraphs.
It would be helpful if there was also terminology for the vertex sets $U$ (with maximal degree 1) and $V$ (with arbitrary degree).
Background: the application is in parallel computing where each vertex has a canonical owner, but may be "ghosted" in other memory spaces. Sometimes the term "local-to-global map" is used for this, but the concept is more general.