Unavoidable Minors of Large 3-Connected Matroids

Ding, Guoli; Oporowski, Bogdan; Oxley, James; Vertigan, Dirk

doi:10.1006/jctb.1997.1785

Cited by 35 publications

(67 citation statements)

References 4 publications

(30 reference statements)

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“…The next result [12] extends Theorem 11.5 to arbitrary 3-connected matroids. Its proof, which required the development of some new tools, is outlined in [40].…”

Section: Unavoidability Revisitedsupporting

confidence: 57%

On the interplay between graphs and matroids

Oxley¹

2001

Surveys in Combinatorics, 2001

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“…The next result [12] extends Theorem 11.5 to arbitrary 3-connected matroids. Its proof, which required the development of some new tools, is outlined in [40].…”

Section: Unavoidability Revisitedsupporting

confidence: 57%

On the interplay between graphs and matroids

Oxley¹

2001

Surveys in Combinatorics, 2001

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“…By Theorem 2, 5 Ž . 8.3 iii , U is a strong universal stabilizer for N N. Clearly, no matroid in N N 2,5 is strictly freer than U . Thus, by Theorem 7.4, M has no U -minor.…”

Section: žmentioning

confidence: 97%

“…11.2.16 , since MЈ has U as a 2,5 minor, MЈ has U as a minor. Then, by dualizing the argument above, we 3,5 deduce that M has no U -minor.…”

Section: žmentioning

confidence: 98%

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Weak Maps and Stabilizers of Classes of Matroids

Geelen

Oxley

Vertigan

et al. 1998

Advances in Applied Mathematics

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Let F be a field and let N be a matroid in a class N N of F-representable matroids that is closed under minors and the taking of duals. Then N is an F-stabilizer for N N if every representation of a 3-connected member of N N is determined up to elementary row operations and column scaling by a representation of any one of its N-minors. The study of stabilizers was initiated by Whittle. This paper extends that study by examining certain types of stabilizers and considering the connection with weak maps.The notion of a universal stabilizer is introduced to identify the underlying matroid structure that guarantees that N will be an FЈ-stabilizer for N N for every field FЈ over which members of N N are representable. It is shown that, just as with F-stabilizers, one can establish whether or not N is a universal stabilizer for N N by an elementary finite check. If N is a universal stabilizer for N N, we determine additional conditions on N and N N that ensure that if N is not a strict rank-preserv-*E-mail: jfgeelen@jeeves.uwaterloo.ca. † E-mail: oxley@math.lsu.edu. ‡ E-mail: vertigan@math.lsu.edu. § E-mail: whittle@kauri.vuw.ac.nz.

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“…Ding et al [5] identified certain rank-r 3-connected matroids as being unavoidable in the sense that every sufficiently large 3-connected matroid has one of the specified matroids as a minor. Included among these unavoidable matroids are the wheels and whirls, whose fundamental role within the class of 3-connected matroids is well known.…”

Section: Introductionmentioning

confidence: 99%

On Extremal Connectivity Properties of Unavoidable Matroids

1999

Journal of Combinatorial Theory, Series B

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Unavoidable Minors of Large 3-Connected Matroids

Cited by 35 publications

References 4 publications

On the interplay between graphs and matroids

On the interplay between graphs and matroids

Weak Maps and Stabilizers of Classes of Matroids

On Extremal Connectivity Properties of Unavoidable Matroids

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