Type of Document 
Dissertation 
Author 
Aikin, Jeremy M.

URN 
etd07082009102124 
Title 
The Structure of 4Separations in 4Connected 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
 2equivalence

Date of Defense 
20090612 
Availability 
unrestricted 
Abstract
Oxley, Semple and Whittle described a tree decomposition for a 3connected matroid M that displays, up to a natural equivalence, all nontrivial 3separations of M. Crossing 3separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a kflower, with no assumptions on the connectivity of M. We completely classify kflowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4connected matroids, we give a new notion of equivalence of 4separations that we show will be needed to describe a tree decomposition for 4connected matroids. Finally, we characterize all internally 4connected 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 4separating. 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 singleelement extension of a wheel, or a particular singleelement extension of the bond matroid of a cubic ladder.

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