Carolyn Chun

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tuttes Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)-E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)-E(N)|=<3 unless M or its dual is the cycle matroid of a planar or Mobius quartic ladder, or a 16-element variant of such a planar ladder.

UNAVOIDABLE PARALLEL MINORS OF REGULAR MATROIDS

To the memory of Tom Brylawski, who contributed so much to matroid theory. Abstract. We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M � (K3,k), M(Wk), M(Kk), the cycle matroid of the graph obtained from K2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,k by adding a complete graph on the vertex class with three vertices. For 3-connected graphs, the collections of unavoidable parallel and unavoidable series minors were determined by Chun, Ding, Oporowski, and Vertigan (3) and by Oporowski, Oxley, and Thomas (8). In this paper, we combine these results with Seymours decomposition theorem for regular matroids (12) to determine the collection of unavoidable parallel minors for the class of 3-connected regular ma- troids. In particular, we prove that the last collection is precisely the union of the collections of unavoidable parallel minors for the classes of 3-connected graphic and 3-connected cographic matroids. The collections of unavoidable minors for binary 3-connected matroids and for all 3-connected matroids were determined in (6, 7). From the first of these, one can determine the collection of unavoidable minors for regular 3-connected matroids, although this result had been obtained earlier by Ding and Oporowski (5). We would like to extend our main theorem to find the unavoidable parallel minors for the class of binary 3-connected matroids, but this will require some new ideas. Our terminology for matroids and graphs generally follows (9) and (4). If M and N are both matroids or are both graphs, N is a parallel minor of M if N can be obtained from M by a sequence of moves each consisting of contracting an element (in the graph case, an edge) or deleting an element that is in a 2-element circuit. When M and N are both matroids, N is a series minor of M if N ∗ is a parallel minor of M ∗ . If G and H are graphs and H is a parallel minor of G, then M(H) is a parallel minor of M(G). Conversely, when G and H are loopless 3-connected graphs, if M(H) is a parallel minor of M(G), then H is a parallel minor of G. Let M be a matroid with ground set E and rank function r. The simplification of M will be denoted by si(M). The connectivity functionM of M is defined for all subsets X of E byM(X) = r(X) + r(E − X) − r(M). Equivalently, �M(X) = r(X) + r ∗ (X) − |X|. ThusM(X) = �M ∗(X). For a positive integer m, whenM(X) < m, a partition (X,Y ) of E is an m-separation if min{|X|, |Y |} ≥ m and is a vertical m-separation if min{r(X),r(Y )} ≥ m. A matroid is n-connected

Towards a splitter theorem for internally 4-connected binary matroids IX

Let M be a binary matroid that is internally 4-connected, that is, M is 3-connected, and one side of every 3-separation is a triangle or a triad. Let N be an internally 4-connected proper minor of M. In this paper, we show that M has a proper internally 4-connected minor with an N-minor that can be obtained from M either by removing at most three elements, or by removing some set of elements in an easily described way from one of a small collection of special substructures of M.

On the interplay between embedded graphs and delta-matroids: EMBEDDED GRAPHS AND DELTA-MATROIDS

The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections between geometric duals of plane graphs and duals of matroids. We obtain analogous connections for various types of duality in the literature for graphs in surfaces of higher genus and delta-matroids. Using this interplay, we establish a rough structure theorem for delta-matroids that are twists of matroids, we translate Petrie duality on ribbon graphs to loop complementation on delta-matroids, and we prove that ribbon graph polynomials, such as the Penrose polynomial, the characteristic polynomial, and the transition polynomial, are in fact delta-matroidal. We also express the Penrose polynomial as a sum of characteristic polynomials.

Inductive tools for connected delta-matroids and multimatroids

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.

On zeros of the characteristic polynomial of matroids of bounded tree-width

Abstract We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power q and constant k , the characteristic polynomial of any loopless, G F ( q ) -representable matroid with tree-width k has no real zero greater than q k − 1 .

Unavoidable Connected Matroids Retaining a Specified Minor

A sufficiently large connected matroid

Matroids, Delta-matroids and Embedded Graphs

M

Towards a splitter theorem for internally 4-connected binary matroids

contains a big circuit or a big cocircuit. Wu showed that we can ensure that

On the interplay between embedded graphs and delta-matroids

M