Rhiannon Hall

On matroids of branch-width three

For all positive integers k, the class Bk of matroids of branch-width at most k is minor-closed. When k is 1 or 2, the class Bk is, respectively, the class of direct sums of loops and coloops, and the class of direct sums of series-parallel networks. B3 is a much richer class as it contains infinite antichains of matroids and is thus not well-quasi-ordered under the minor order. In this paper, it is shown that, like B1 and B2, the class B3 can be characterized by a finite list of excluded minors.

A chain theorem for 4-connected matroids

A matroid M is said to be k-connected up to separators of size l if whenever A is (k - 1)-separating in M, then either |A| ≤ l or |E(M) - A|≤ l. We use si(M) and co(M) to denote the simplification and cosimplification of the matroid M. We prove that if a 3-connected matroid M is 4-connected up to separators of size 5, then there is an element x of M such that either co(M\x) or si(M/x) is 3-connected and 4-connected up to separators of size 5, and has a cardinality of |E(M)| - 1 or |E(M)| - 2.

Fork-decompositions of matroids

One of the central problems in matroid theory is Rotas conjecture that, for all prime powers q, the class of GF(q)-representable matroids has a finite set of excluded minors. This conjecture has been settled for q= 5, there are 3-connected GF(q)-representable matroids having arbitrarily many inequivalent GF(q)-representations. This fact refutes a 1988 conjecture of Kahn that 3-connectivity would be strong enough to ensure an absolute bound on the number of such inequivalent representations. This paper introduces fork-connectivity, a new type of self-dual 4-connectivity, which we conjecture is strong enough to guarantee the existence of such a bound but weak enough to allow for an analogue of Seymours Splitter Theorem. We prove that every fork-connected matroid can be reduced to a vertically 4-connected matroid by a sequence of operations that generalize @D-Y and Y-@D exchanges. It follows from this that the analogue of Kahns Conjecture holds for fork-connected matroids if and only if it holds for vertically 4-connected matroids. The class of fork-connected matroids includes the class of 3-connected forked matroids. By taking direct sums and 2-sums of matroids in the latter class, we get the class M of forked matroids, which is closed under duality and minors. The class M is a natural subclass of the class of matroids of branch-width at most 3 and includes the matroids of path-width at most 3. We give a constructive characterization of the members of M and prove that M has finitely many excluded minors.

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 .