Abstract.
An edge e of a simple 3-connected graph G is essential if neither the deletion G\e nor the contraction G/e is both simple and 3-connected. Tutte’s Wheels Theorem proves that the only simple 3-connected graphs with no non-essential edges are the wheels. In earlier work, as a corollary of a matroid result, the authors determined all simple3-connected graphs with at most two non-essential edges. This paper specifies all such graphs with exactly three non-essential edges. As a consequence, with the exception of the members of 10 classes of graphs, all 3-connected graphs have at least four non-essential edges.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Acknowledgments The first author was partially supported by grants from the National Security Agency. The second author was partially supported by the Office of Naval Research under Grant No. N00014-01-1-0917.
1991 Mathematics Subject Classification: 05C40
Final version received: October 30, 2003
Rights and permissions
About this article
Cite this article
Oxley, J., Wu, H. The 3-Connected Graphs with Exactly Three Non-Essential Edges. Graphs and Combinatorics 20, 233–246 (2004). https://doi.org/10.1007/s00373-004-0552-5
Received:
Issue Date:
DOI: https://doi.org/10.1007/s00373-004-0552-5