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

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 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 structure of k-connected graphs without Kk-minor

Suppose G is a k-connected graph that does not contain Kk as a minor. What does G look like? This question is motivated by Hadwiger’s conjecture (Vierteljahrsschr. Naturforsch. Ges. Zürich 88 (1943) 133) and a deep result of Robertson and Seymour (J. Combin. Theory Ser. B. 89 (2003) 43). It is easy to see that such a graph cannot contain a (k−1)-clique, but could contain a (k−2)-clique, as Kk−5...

On minor-minimally-connected matroids

Exponential Speedup of Fixed Parameter Algorithms on K_{3,3}-minor-free or K_5-minor-free Graphs