Partitioning a bipartite graph to get disjoint components of same size I have a bipartite graph $G = (V, E)$ where $V = S \cup T$ is the division into the two halves. I want to select $n$ elements from $S$ and $nk$ elements from $T$ such that the graph they generate has $n$ connected components, each of size $k+1$ and containing one element of $S$. To be clear, $n$ and $k$ are inputs to the problem: it's not a maximisation problem.

The naïve brute force algorithm would consider $\binom{S}{n}$ candidate subsets of $S$, and would do $O(E)$ work for each to identify vertices in $T$ which have an edge to exactly one element of the candidate subset of $S$. For fixed $n$ that's $O(S^n E)$.

I'm not optimistic, because it's an independence problem, but is there an algorithm for fixed $n$ and $k$ whose running time is polynomial in $E$ with exponent independent of $n$ and $k$?
Your problem is NP-hard, even for any fixed $k \geq 3$ by a straight-forward reduction from 3-SET-COVER. Quick explanation: nodes of $S$ are triplets, nodes of $T$ are set elements and each triplet connected to all elements it contains.

I am still thinking about the case when $n$ is fixed.