On the minor-minimal 2-connected graphs having a fixed minor Original Research Article

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 fails.