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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
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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

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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