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University of Canterbury. Mathematics and Statistics.
Doi
Abstract
One of the central problems in matroid theory is Rota's 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 ≤ 4 but
remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all 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 Seymour's
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
Δ − Y and Y − Δ exchanges. It follows from this that the analogue of Kahn's
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
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