Fork-decompositions of matroids (2004)
AuthorsHall, R., Oxley, J., Semple, C., Whittle, G.show all
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.
CitationHall, R., Oxley, J., Semple, C., Whittle, G. (2004) Fork-decompositions of matroids. Advances in Applied Mathematics, 32, pp. 523-575.
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