On the minor-minimal 2-connected graphs having a fixed minor

Let H be a graph with κ1 components and κ2 blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)| − |E(H)| ≤ α(κ1 − 1) + β(κ2 − 1) for all (α, β) such that α + β ≥ 5, 2α + 5β ≥ 20, and β ≥ 3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality