The structure of the 3-separations of 3-connected matroids |
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| Authors: | James Oxley Charles Semple Geoff Whittle |
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| Institution: | a Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA;b Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand;c School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand |
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| Abstract: | Tutte defined a k-separation of a matroid M to be a partition (A,B) of the ground set of M such that |A|,|B|k and r(A)+r(B)−r(M)<k. If, for all m<n, the matroid M has no m-separations, then M is n-connected. Earlier, Whitney showed that (A,B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M. |
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| Keywords: | 3-Connected matroid Tree decomposition 3-Separation Tutte connectivity |
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