Abstract.
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.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: December 24, 1996 Revised: September 21, 1998
Rights and permissions
About this article
Cite this article
Oxley, J., Wu, H. Matroids and Graphs with Few Non-Essential Elements. Graphs Comb 16, 199–229 (2000). https://doi.org/10.1007/PL00021178
Issue Date:
DOI: https://doi.org/10.1007/PL00021178