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Full-Text Articles in Physical Sciences and Mathematics
On Matroid And Polymatroid Connectivity, Dennis Wayne Hall Ii
On Matroid And Polymatroid Connectivity, Dennis Wayne Hall Ii
LSU Doctoral Dissertations
Matroids were introduced in 1935 by Hassler Whitney to provide a way to abstractly capture the dependence properties common to graphs and matrices. One important class of matroids arises by taking as objects some finite collection of one-dimensional subspaces of a vector space. If, instead, one takes as objects some finite collection of subspaces of dimensions at most k in a vector space, one gets an example of a k-polymatroid.
Connectivity is a pivotal topic of study in the endeavor to understand the structure of matroids and polymatroids. In this dissertation, we study the notion of connectivity from several …
The Structure Of 4-Separations In 4-Connected Matroids, Jeremy M. Aikin
The Structure Of 4-Separations In 4-Connected Matroids, Jeremy M. Aikin
LSU Doctoral Dissertations
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 …