Identifier

etd-07082009-102124

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we characterize all internally 4-connected binary matroids M with the property that the ground set of M can be cyclically ordered so that any consecutive collection of elements in this cyclic ordering is 4-separating. We prove that in this case either M is a matroid on at most seven elements or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder.

Date

2009

Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Committee Chair

Oxley, James G.

DOI

10.31390/gradschool_dissertations.898

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