Extensions of Tutte\u27s wheels-and-whirls theorem

Tutte\u27s wheels-and-whirls theorem states that if M is a 3-connected matroid and, for every element e, both the deletion and the contraction of e destroy 3-connectivity, then M is a wheel or a whirl. We prove some extensions of this theorem, one of which states that if M is 3-connected and has both a wheel and a whirl minor, then either M has only seven elements or there is some element the deletion or contraction of which maintains 3-connectivity and leaves a matroid with both a wheel and a whirl minor. © 1992

The Determination of a Matroid\u27s Structure From Properties of Certain Large Minors.

This dissertation solves some problems related to the structure of matroids. In Chapter 2, we prove that if M and N are distinct connected matroids on a common ground set E, where $\vert E\vert \ge 2,$ and, for every e in $E,\ M\\ e = N\\ e$ or M/e = N/e, then one of M and N is a relaxation of the other. In addition, we determine the matroids M and N on a common ground set E such that, for every pair of elements $\{ e,f\}$ of E, at least two of the four corresponding minors of M and N obtained by eliminating e and f are equal. The theorems in Chapter 3 and 4 extend a result of Oxley that characterizes the non-binary matroids M such that, for each element e, $M\\ e$ or M/e is binary. In Chapter 3, we describe the non-binary matroids M such that, for every pair of elements $\{ e,f\} .$ at least two of the four minors of M obtained by eliminating e and f are binary. In Chapter 4, we obtain an alternative extension of Oxley\u27s result by changing the minor under consideration from the smallest 3-connected whirl, U\sb{2,4}, to the smallest 3-connected wheel, M(K\sb4). In particular, we determine the binary matroids M having an M(K\sb4)-minor such that, for every element e, $M\\ e$ or M/e has no M(K\sb4)-minor. This enables us to characterize the matroids M that are not series-parallel networks, but, for every $e,\ M\\ e$ or M/e is a series-parallel network