Computing excluded minors for classes of matroids representable over partial fields

We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids

NN-detachable pairs in 3-connected matroids II: life in XX

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the second in a series of three papers where we describe the structures that arise when it is not possible to find an $N$-detachable pair in $M$. In the first paper in the series, we showed that if $M$ has no $N$-detachable pairs, then either $M$ has one of three particular 3-separators that can appear in a matroid with no $N$-detachable pairs, or there is a 3-separating set $X$ with certain strong structural properties. In this paper, we analyse matroids with such a structured set $X$, and prove that they have either an $N$-detachable pair, or one of five particular 3-separators that can appear in a matroid with no $N$-detachable pairs.Comment: 47 pages, 5 figure

The excluded minors for 2- and 3-regular matroids

The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of matroids representable over the Hydra-5 partial field, and the 3-connected matroids in the class with a $U_{2,5}$- or $U_{3,5}$-minor are precisely those with six inequivalent representations over GF(5). We also prove that an excluded minor for this class has at most 15 elements.Comment: 79 pages, 1 figur