Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-16T07:04:03.954Z Has data issue: false hasContentIssue false
  • English
  • Français

On 3-Connected Matroids

Published online by Cambridge University Press:  20 November 2018

James G. Oxley
James G. Oxley*
Affiliation:
Australian National University, Canberra, Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper extends several graph-theoretic results to matroids. The main result of Tutte's paper [10] which introduced the theory of n-connection for matroids was a generalization of an earlier result of his [9] for 3-connected graphs. The latter has since been strengthened by Halin [3] and in Section 3 of this paper we prove a matroid analogue of Halin's result. Tutte used his result for 3-connected graphs to deduce a recursive construction of all simple 3-connected graphs having at least four vertices. In Section 4 we generalize this by giving a recursive construction of all 3-connected matroids of rank at least three. Section 2 contains a generalization to minimally n-connected matroids of a result of Dirac [2] for minimally 2-connected graphs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 20 - 27
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bondy, J. A. and Murty, U. S. R., Graph theory with applications (Macmillan, London ; American Elsevier, New York, 1976).Google Scholar
2. Dirac, G. A., Minimally 2–connected graphs, J. Reine Angew. Math. 228 (1967), 204–216.Google Scholar
3. Halin, R., Zur Théorie der n-fach zusammenhangenden Graphen, Abh. Math. Sem. Univ. Hamburg 33 (1969), 133–164.Google Scholar
4. Inukai, T. and Weinberg, L., Theorems on matroid connectivity, Discrete Math. 22 (1978), 311–312.Google Scholar
5. Oxley, J. G., On matroid connectivity (submitted).Google Scholar
6. Oxley, J. G., On a matroid generalization of graph connectivity, (submitted).Google Scholar
7. Richardson, W. R. H., Decomposition of chain-groups and binary matroids, Proc. Fourth South-Eastern Conf. on Combinatorics, Graph Theory, and Computing (Utilitas Mathematica, Winnipeg, 1973), 463–476.Google Scholar
8. Seymour, P. D., Decomposition of regular malroids, J. Combin. Theory Ser. B (to appear).Google Scholar
9. Tutte, W. T., A theory of 3–connected graphs, Nederl. Akad. Wetensch. Proc. Ser. A 64 (1961), 441–455.Google Scholar
10. Tutte, W. T., Connectivity in matroids, Can. J. Math. 18 (1966), 1301–1324.Google Scholar
11. Tutte, W. T., Wheels and whirls, in Théorie des matroides (Lecture Notes in Mathematics Vol. 211, Springer-Verlag, Berlin, Heidelberg, New York, 1971), 1–4.Google Scholar
12. Welsh, D. J. A., Matroid theory (Academic Press, London, New York, San Francisco, 1976).Google Scholar
13. Wong, P.-K., On certain n-connected matroids, J. Reine Angew. Math. 299/300 (1978), 1–6.Google Scholar