Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T17:26:45.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Local Extremal Connectivity Results for Matroids

Published online by Cambridge University Press:  12 September 2008

Safwan Akkari  and
James Oxley
Safwan Akkari
Affiliation:
Department of Mathematical Sciences, Indiana University–Purdue University at Fort Wayne, Fort Wayne, Indiana 46805-1499
James Oxley
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918
Get access Rights & Permissions [Opens in a new window]

Abstract

Tutte proved that if e is an element of a 3-connected matroid M such that neither M\e nor M/e is 3-connected, then e is in a 3-circuit or a 3-cocircuit. In this paper, we prove a broad generalization of this result. Among the consequences of this generalization are that if X is an (n − 1)-element subset of an n-connected matroid M such that neither M\X nor M/X is connected, then, provided |E(M)| ≥ 2(n − 1)≥ 4, X is in both an n-element circuit and an n-element cocircuit. When n = 3, we describe the structure of M more closely using Δ − Y exchanges. Several related results are proved and we also show that, for all fields F other than GF(2), the set of excluded minors for F-representability is closed under both Δ − Y and Y − Δ exchanges.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 367 - 384
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Akkari, S. (1992) A minimal 3-connectedness result for matroids. Discrete Math. 103 221–232.CrossRefGoogle Scholar
[2]Akkari, S. and Oxley, J. (1991) Some extremal connectivity results for matroids. J. Combin. Theory Ser. B 52 301–320.CrossRefGoogle Scholar
[3]Bixby, R. E. (1979) On Reid's characterization of the ternary matroids. J. Combin. Theory Ser. B 26 174–204.CrossRefGoogle Scholar
[4]Bixby, R. E. (1982) A simple theorem on 3-connectivity. Linear Algebra Appl. 45 123–126.Google Scholar
[5]Bixby, R. E. and Coullard, C. R. (1987) Finding a smallest 3-connected minor maintaining a fixed minor and a fixed element. Combinatorica 7 231–242.Google Scholar
[6]Brylawski, T. H. (1971) A combinatorial model for series-parallel networks. Trans. Amer. Math. Soc. 154 1–22.CrossRefGoogle Scholar
[7]Brylawski, T. H. (1975) Modular constructions for combinatorial geometries. Trans. Amer. Math. Soc. 203 1–14.Google Scholar
[8]Brylawski, T. H. (1986) Constructions. In: White, N. (ed.) Theory of Matroids, Cambridge University Press, Cambridge127–223.CrossRefGoogle Scholar
[9]Crapo, H. H. (1967) Structure theory for geometric lattices. Rend. Sem. Mat. Univ. Padova 38 14–22.Google Scholar
[10]Cunningham, W. H. (1981) On matroid connectivity. J. Combin. Theory Ser. B 30 94–99.CrossRefGoogle Scholar
[11]Higgs, D. A. (1968) Strong maps of geometries. J. Combin. Theory 5 185–191.CrossRefGoogle Scholar
[12]Inukai, T. and Weinberg, L. (1978) Theorems on matroid connectivity. Discrete Math. 22 311–312.Google Scholar
[13]Inukai, T. and Weinberg, L. (1981) Whitney connectivity of matroids. SIAM J. Alg. Disc. Methods 2 108–120.CrossRefGoogle Scholar
[14]Lemos, M. (1989) On 3-connected matroids. Discrete Math. 73 273–283.CrossRefGoogle Scholar
[15]Murty, U. S. R. (1974) Extremal critically connected matroids. Discrete Math. 8 49–58.CrossRefGoogle Scholar
[16]Oxley, J. G. (1981) On connectivity in matroids and graphs. Trans. Amer. Math. Soc. 265 47–58.CrossRefGoogle Scholar
[17]Oxley, J. G. (1981) On matroid connectivity. Quart. J. Math. Oxford Ser. (2) 32 193–208.Google Scholar
[18]Oxley, J. G. (1981) On 3-connected matroids. Canad. J. Math. 33 20–27.CrossRefGoogle Scholar
[19]Oxley, J. G. (1981) On a matroid generalization of graph connectivity. Math. Proc. Camb. Phil. Soc. 90 207–214.CrossRefGoogle Scholar
[20]Oxley, J. G. (1984) On minor-minimally-connected matroids. Discrete Math. 51 63–72.CrossRefGoogle Scholar
[21]Oxley, J. G. (1986) On the matroids representable over GF(4). J. Combin. Theory Ser. B 41 250–252.Google Scholar
[22]Oxley, J. G. (1992) Matroid Theory, Oxford University Press, New York.Google Scholar
[23]Richardson, W. R. H. (1973) Decomposition of chain-groups and binary matroids. Proc. Fourth Southeastern Conf. on Combinatorics, Graph Theory and Computing, Utilitas Mathematica, Winnipeg463–476.Google Scholar
[24]Robertson, N. and Seymour, P. D. (1984) Generalizing Kuratowski's Theorem. Congressus Numerantium 45 129–138.Google Scholar
[25]Seymour, P. D. (1979) Matroid representation over GF(3). J. Combin. Theory Ser. B 26 159–173.CrossRefGoogle Scholar
[26]Seymour, P. D. (1980) Decomposition of regular matroids. J. Combin. Theory Ser. B 28 305–359.CrossRefGoogle Scholar
[27]Truemper, K. (1992) Matroid Decomposition, Academic Press, New York.Google Scholar
[28]Tutte, W. T. (1966) Connectivity in matroids. Canad. J. Math. 18 1301–1324.CrossRefGoogle Scholar
[29]Wong, P.-K. (1978) On certain n-connected matroids. J. Reine Angew. Math. 299/300 1–6.Google Scholar