  It is well-known that the Vertex Cover problem is in P on bipartite graphs,
however; the computational complexity of the Partial Vertex Cover problem on
bipartite graphs is open. In this paper, we first show that the Partial Vertex
Cover problem is NP-hard on bipartite graphs. We then identify an interesting
special case of bipartite graphs, for which the Partial Vertex Cover problem
can be solved in polynomial-time. We also show that the set of acyclic
bipartite graphs, i.e., forests, and the set of bipartite graph where the
degree of each vertex is at most 3 fall into that special case. Therefore, we
prove that the Partial Vertex Cover problem is in P on trees, and it is also in
P on the set of bipartite graphs where the degree of each vertex is at most 3.
