I have this problem, which I haven't seen before in the literature: given a bipartite graph and a natural number $k$, can we select at least $k$ of the edges such that each left-hand vertex is incident to exactly two or exactly zero selected edges, and each right-hand vertex is incident to at most one selected edge?
Obviously if replace "two" with one we have the matching problem, which can be solved in polynomial time. I've tried in vain to solve my problem using network flow or matching, and am also stumped when I try to reduce from NP-complete problems.