On the structure of 3-connected matroids and graphs
An element \(e\) of a 3-connected matroid \(M\) is essential if deleting \(e\) from \(M\), as well as contracting \(e\), destroys 3-connectivity. \textit{W. T. Tutte} [Can. J. Math. 18, 1301-1324 (1966; Zbl 0149.21501)] showed that the only 3-connected matroids in which every element is essential are wheels and whirls. In the paper under review the authors begin to explore the structure of matroids and graphs in which the set of non-essential elements is small but not empty. It is shown that every essential element in a 3-connected matroid \(M\) (that is not a wheel or a whirl) is contained in a partial wheel, called a fan, both ends of which are non-essential, i.e. if \(M\) contains non-essential elements, there must be at least two of them. Moreover, essential elements can be partitioned into classes, where two elements are in the same class if \(M\) has a fan containing both. The second main result of the paper says that a matroid \(M\) containing a fan on \(2k\) or \(2k+1\) elements, where \(k\) is at least 2, can be obtained by sticking together a \((k+1)\)-spoked wheel and a certain 3-connected minor of \(M\). These results are used by the authors in another paper [Graphs Comb. 16, No 2, 199-229 (2000; Zbl 0966.05016)] to completely determine all 3-connected matroids with exactly two non-essential elements.
scientific article; zbMATH DE number 1512039
scientific article