On the Structure of 3-connected Matroids and Graphs

@article{Oxley2000OnTS,
  title={On the Structure of 3-connected Matroids and Graphs},
  author={James G. Oxley and Haidong Wu},
  journal={Eur. J. Comb.},
  year={2000},
  volume={21},
  pages={667-688}
}
An element e of a 3 -connected matroid M is essential if neither the deletionM\e nor the contraction M/e is 3 -connected. Tutte?s Wheels and Whirls Theorem proves that the only 3 -connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3 -connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into… 
Matroids and Graphs with Few Non-Essential Elements
TLDR
Tutte's Wheels and Whirls Theorem proves that the only 3- connected matroids in which every element is essential are the wheels and whirls, and it is proved that every 3-connected matroid M for which no single-element contraction is3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3- Connected.
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  • Mathematics
    Combinatorics, Probability and Computing
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TLDR
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The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s
Totally Free Expansions of Matroids
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References

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Matroids and Graphs with Few Non-Essential Elements
TLDR
Tutte's Wheels and Whirls Theorem proves that the only 3- connected matroids in which every element is essential are the wheels and whirls, and it is proved that every 3-connected matroid M for which no single-element contraction is3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3- Connected.
Some Local Extremal Connectivity Results for Matroids
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
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It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid
Some extremal connectivity results for matroids
Connectivity in Matroids
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  • Mathematics
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  • 1966
An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only
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