How many vertices should be disconnected to make a graph acyclic?
A set of vertices whose removal makes the graph acyclic is called a feedback vertex set (FVS). What you seem to be doing is finding the smallest FVS, subdividing all edges incident to each vertex in the FVS, and finally deleting the vertices in the FVS.
Computing the smallest FVS is NP-hard, but it admits 2-approximation.