Cyclic flats of binary matroids

We generalize a minimal 3-connectivity result of Halin from graphs to binary matroids. As applications of this theorem to minimally 3-connected matroids, we obtain new results and short inductive proofs of results of Oxley and Wu. We also give new short inductive proofs of results of Dirac and Halin on minimally k-connected graphs for k ∈ {2,3}.

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A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs

Mathematical Programming ◽

10.1007/s10107-010-0425-z ◽

2010 ◽

Vol 133 (1-2) ◽

pp. 203-225 ◽

Cited By ~ 11

Author(s):

João Gouveia ◽

Monique Laurent ◽

Pablo A. Parrilo ◽

Rekha Thomas

Keyword(s):

Semidefinite Programming ◽

Binary Matroids

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The Penrose Polynomial of Binary Matroids

Monatshefte für Mathematik ◽

10.1007/s006050070020 ◽

2000 ◽

Vol 131 (1) ◽

pp. 1-13 ◽

Cited By ~ 6

Author(s):

Martin Aigner ◽

Hans Mielke

Keyword(s):

Binary Matroids

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Binary Matroids

Combinatorial Geometries ◽

10.1017/cbo9781107325715.004 ◽

1987 ◽

pp. 28-39 ◽

Cited By ~ 1

Author(s):

J.C. Fournier

Keyword(s):

Binary Matroids

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Dependencies Among Dependencies in Matroids

The Electronic Journal of Combinatorics ◽

10.37236/8742 ◽

2019 ◽

Vol 26 (3) ◽

Author(s):

James Oxley ◽

Suijie Wang

Keyword(s):

Binary Matroids

In 1971, Rota introduced the concept of derived matroids to investigate "dependencies among dependencies" in matroids. In this paper, we study the derived matroid $\delta M$ of an ${\mathbb F}$-representation of a matroid $M$. The matroid $\delta M$ has a naturally associated ${\mathbb F}$-representation, so we can define a sequence $\delta M$, $\delta^2 M$, \dots . The main result classifies such derived sequences of matroids into three types: finite, cyclic, and divergent. For the first two types, we obtain complete characterizations and thereby resolve some of the questions that Longyear posed in 1980 for binary matroids. For the last type, the divergence is estimated by the coranks of the matroids in the derived sequence.

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