# On the Structure of 3-connected Matroids and Graphs

@article{Oxley2000OnTS, title={On the Structure of 3-connected Matroids and Graphs}, author={James G. Oxley and Haidong Wu}, journal={Eur. J. Comb.}, year={2000}, volume={21}, pages={667-688} }

An element e of a 3 -connected matroid M is essential if neither the deletionM\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 at least two such elements. We prove that the essential elements of M can be partitioned into…

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## 46 Citations

The structure of a 3-connected matroid with a 3-separating set of essential elements

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