Triangles in 3-connected matroids

AbstractA collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S8 or the generalized parallel connection across T of F7 and M(K4)

The Smallest Rounded Sets of Binary Matroids

It was proved by Oxley that U2,4 is the only non-trivial 3-connected matroid N such that, whenever a 3-connected matroid M has an N-minor and x and y are elements of M, there is an N-minor of M using {x, y} . This paper establishes the corresponding result for binary matroids by proving that if M and N above must both be binary, then there are exactly two possibilities for N: the rank-three and rank-four wheels. © 1990, Academic Press Limited. All rights reserved

On Roundedness in Matroid Theory.

This thesis studies the relationship between subsets and specified minors in a 3-connected matroid. For positive integers k and m, a set S of k-connected matroids is (k,m)-rounded if it satisfies the following condition. Whenever M is a k-connected matroid having an S-minor and X is a subset of E(M) with at most m elements, then M has an S-minor using X. Oxley characterized the (3,2)-rounded sets that contain a single matroid. In Chapter 2, we obtain an analog of this result for binary matroids. In Chapter 3, we use this result to characterize the pairs of matroids which form (3,2)-rounded sets. The methods of Chapter 3 are generalized to 4-connected matroids in Chapter 4 to determine the (4,2)-rounded sets that contain a single matroid. This extends results of Coullard and Kahn. For a 3-connected minor N of a 3-connected matroid M, the following question arises from roundedness theory. Let X be a subset of E(M). How small a 3-connected minor of M can we find which both uses X and has an N-minor? Seymour answered this question for $\vert$X$\vert$ = 1 and 2. We answer this question for $\vert$X$\vert$ $\geq$ 3 in Chapter 5. Finally, in Chapter 6, results from roundedness theory are applied to the study of 3-element circuits in 3-connected matroids. An extension of a result of Asano, Nishizeki, and Seymour is obtained for binary matroids which are non-regular

On deletions of largest bonds in graphs

AbstractA well-known conjecture of Scott Smith is that any two distinct longest cycles of a k-connected graph must meet in at least k vertices when k≥2. We provide a dual version of this conjecture for two distinct largest bonds in a graph. This dual conjecture is established for k⩽6

On the circuit-spectrum of binary matroids

AbstractMurty, in 1971, characterized the connected binary matroids with all circuits having the same size. We characterize the connected binary matroids with circuits of two different sizes, where the largest size is odd. As a consequence of this result we obtain both Murty’s result and other results on binary matroids with circuits of only two sizes. We also show that it will be difficult to complete the general case of this problem

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

Let N be a restriction of a 3-connected matroid M and let M′ be a 3-connected minor of M that is minimal having N as a restriction. This paper gives a best-possible upper bound on \E(M′)- E(N)\. © Springer-Verlag 2000

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

Let N be a restriction of a 3-connected matroid M and let M' be a 3-connected minor of M that is minimal having N as a restriction. This paper gives a best-possible upper bound on |E(M') - E(N)|