| 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.
|
| Files |
| Filename |
Size |
Approximate Download Time
(Hours:Minutes:Seconds)
|
| 28.8 Modem |
56K Modem |
ISDN (64 Kb) |
ISDN (128 Kb) |
Higher-speed Access |
| |
Aikin_diss.pdf |
9.79 Mb |
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00:23:19 |
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