Title page for ETD etd-07082009-102124


Type of Document Dissertation
Author Aikin, Jeremy M.
URN etd-07082009-102124
Title The Structure of 4-Separations in 4-Connected Matroids
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Oxley, James G. Committee Chair
Perlis, Robert Committee Member
Baldridge, Scott Committee Member
Oporowski, Bogdan Committee Member
Shipman, Stephen Committee Member
Trahan, Jerry Dean's Representative
Keywords
  • matroid
  • connectivity
  • separation
  • flowers
  • local connectivity
  • pods
  • 2-equivalence
Date of Defense 2009-06-12
Availability unrestricted
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.
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