Fork-decompositions of matroids

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 ∆ - Y and Y - ∆ 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.