#### Title

Fork-decompositions of matroids

#### Document Type

Article

#### Publication Date

1-1-2004

#### 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. © 2003 Elsevier Inc. All rights reserved.

#### Publication Source (Journal or Book title)

Advances in Applied Mathematics

#### First Page

523

#### Last Page

575

#### Recommended Citation

Hall, R., Oxley, J., Semple, C., & Whittle, G.
(2004). Fork-decompositions of matroids.* Advances in Applied Mathematics**, 32* (3), 523-575.
https://doi.org/10.1016/S0196-8858(03)00058-7