  Finite obstruction sets for lower ideals in the minor order are guaranteed to
exist by the Graph Minor Theorem. It has been known for several years that, in
principle, obstruction sets can be mechanically computed for most natural lower
ideals. In this paper, we describe a general-purpose method for finding
obstructions by using a bounded treewidth (or pathwidth) search. We illustrate
this approach by characterizing certain families of cycle-cover graphs based on
the two well-known problems: $k$-{\sc Feedback Vertex Set} and $k$-{\sc
Feedback Edge Set}. Our search is based on a number of algorithmic strategies
by which large constants can be mitigated, including a randomized strategy for
obtaining proofs of minimality.
