How to optimally seperate a student body?
It sounds like you have properly reduced the maximum bipartite matching problem to the maximum flow problem, all you need is the reasoning to convince yourself why it is true. You are worried about the complement, so let's consider that. Remember that there is a duality among the minimum cut problem and the maximum flow problem so in finding the minimum cut you will equivalently find the maximum flow. For the definition of minimum cut:


  a partition of the vertices of a graph into two disjoint subsets that are joined by at least one edge whose cut set has the smallest number of edges (unweighted case) or smallest sum of weights possible. Several algorithms exist to find minimum cuts.


The minimum cut for the graph you have constructed will have the smallest number of edges since all the weights are uniform and therefore the smallest number of people from the two sets who want to work togethers given that:

$w(p_i, q_j) = 1 \Leftrightarrow$ person $p_i$ does not want to work with person $q_j$