#### 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.

#### Recommended Citation

Aikin, Jeremy M., "The structure of 4-separations in 4-connected matroids" (2009). *LSU Doctoral Dissertations*. 898.

https://digitalcommons.lsu.edu/gradschool_dissertations/898

#### Committee Chair

Oxley, James G.