Michael J. Dinneen, Kevin Cattell, Michael R. Fellows. Forbidden minors to graphs with small feedback sets. Discrete Mathematics, 230(1-3):215-252, 2001. [doi]
Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. In this paper we characterize several families of graphs with small feedback sets, namely $k_1$-{\sc Feedback Vertex Set}, $k_2$-{\sc Feedback Edge Set} and VertexEdgeFamily{k_1}{k_2}, for small integer parameters $k_1$ and $k_2$. Our constructive methods can compute obstruction sets for any minor-closed family of graphs, provided the pathwidth (or treewidth) of the largest obstruction is known.