An upgraded Wheels-and-Whirls Theorem for 3-connected matroids
نویسندگان
چکیده
Let M be a 3-connected matroid that is not a wheel or a whirl. In this paper, we prove that M has an element e such that M\e or M/e is 3-connected and has no 3-separation that is not equivalent to one induced by M .
منابع مشابه
Matroids and Graphs with Few Non-Essential Elements
An essential element of a 3–connected matroid M is one for which neither the deletion nor the contraction is 3–connected. Tutte’s Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3–connected matroid with at least one non-essential element has at least two such element...
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An element e of a 3-connected matroid M is essential if neither the deletion M\e nor the contraction M/e is 3-connected. Tutte’s Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3-connected matroids that have some non-essential elements, showing that every such matroid M must have ...
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Tutte’s Wheels-and-Whirls Theorem is a basic inductive tool for dealing with 3-connected matroids. This paper proves a generalization of that theorem for the class of 2-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. The main result proves that, in a 3-connected 2-polymatroid that is not a whirl or ...
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Tutte’s Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)| − |E(N)| = 1. Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4connected, that is, for every 3-separation (X, Y ) of N , the set E(N) can be ...
متن کاملThe structure of a 3-connected matroid with a 3-separating set of essential elements
An element e of a 3–connected matroid M is essential if neither the deletion nor the contraction of e from M is 3–connected. Tutte’s 1966 Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. It was proved by Oxley and Wu that if a 3–connected matroid M has a non-essential element, then it has at least two such element...
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ورودعنوان ژورنال:
- J. Comb. Theory, Ser. B
دوره 102 شماره
صفحات -
تاریخ انتشار 2012