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 elements. This paper completely determines all 3-connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3-connected matroid M for which no single-element contraction is 3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3-connected. © Springer-Verlag 2000.
Publication Source (Journal or Book title)
Graphs and Combinatorics
Oxley, J., & Wu, H. (2000). Matroids and graphs with few non-essential elements. Graphs and Combinatorics, 16 (2), 199-229. Retrieved from https://digitalcommons.lsu.edu/mathematics_pubs/1245