On the minor-minimal 3-connected matroids having a fixed minor

On the minor-minimal 3-connected matroids having a fixed minor

Let N be a minor of a 3-connected matroid M such that no proper 3-connected minor of M has N as a minor. This paper proves a bound on |E(M)− E(N)| that is sharp when N is connected.

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

On minor-minimally-connected matroids

On the 3-connected matroids that are minimal having a fixed spanning restriction

Let N be a minor of a 3-connected matroid M and let M 0 be a 3-connected minor of M that is minimal having N as a minor. This paper commences the study of the problem of nding a best-possible upper bound on jE(M 0) ? E(N)j. The main result solves this problem in the case that N and M have the same rank.

On the 3-Connected Matroids That are Minimal Having a Fixed Restriction