Computing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats

Eberhardt, Jens Niklas

doi:10.37236/4610

Cited by 9 publications

(12 citation statements)

References 5 publications

(2 reference statements)

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“…The main result in this section, Theorem 7.3, extends a result by Eberhardt [14] for the Tutte polynomial (Theorem 7.1 below) to the G-invariant. We also show that the converse of Theorem 7.3 is false.…”

Section: Configurationssupporting

confidence: 65%

The G-invariant and catenary data of a matroid

Bonin

Kung

2018

Advances in Applied Mathematics

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Section: Configurationssupporting

confidence: 65%

The G-invariant and catenary data of a matroid

Bonin

Kung

2018

Advances in Applied Mathematics

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“…As an example, the matroid from Examples 1 and 2 gets an edge labelling as illustrated in Figure 2: It is clear that this representation is enough to reconstruct the so-called configuration of the matroid, i.e., the isomorphism type of the lattice of cyclic flats, together with the cardinality and rank of the cyclic flats. However, this data does not uniquely determine the matroid, as is shown in [7].…”

Section: Minors Given By Both Restriction and Contractionmentioning

confidence: 96%

On Binary Matroid Minors and Applications to Data Storage over Small Fields

Grezet

Freij-Hollanti

Westerbäck

et al. 2017

Coding Theory and Applications

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Locally repairable codes for distributed storage systems have gained a lot of interest recently, and various constructions can be found in the literature. However, most of the constructions result in either large field sizes and hence too high computational complexity for practical implementation, or in low rates translating into waste of the available storage space. In this paper we address this issue by developing theory towards code existence and design over a given field. This is done via exploiting recently established connections between linear locally repairable codes and matroids, and using matroid-theoretic characterisations of linearity over small fields. In particular, nonexistence can be shown by finding certain forbidden uniform minors within the lattice of cyclic flats. It is shown that the lattice of cyclic flats of binary matroids have additional structure that significantly restricts the possible locality properties of F2-linear storage codes. Moreover, a collection of criteria for detecting uniform minors from the lattice of cyclic flats of a given matroid is given, which is interesting in its own right.The parameters (n, k, d, r, δ) can immediately be defined and studied for matroids in general, as in [18,20]. Matroid fundamentalsMatroids were first introduced by Whitney in 1935, to capture and generalise the notion of linear dependence in purely combinatorial terms.

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“…The configuration of a matroid, which Eberhardt [9] introduced, is obtained from its lattice of cyclic flats (that is, flats that are unions of circuits) by recording the abstract lattice structure along with just the size and rank of each cyclic flat, not the set. Eberhardt proved that from the configuration of a matroid M , one can compute its Tutte polynomial, T (M ; x, y) = A⊆E(M) (x − 1) r(M)−r(A) (y − 1) |A|−r (A) .…”

Section: Introductionmentioning

confidence: 99%

Matroids with different configurations and the same $\mathcal{G}$-invariant

Bonin¹

2021

Preprint

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From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the G-invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have methods for constructing matroids with different configurations but the same G-invariant. We offer several such constructions along with tools for developing more.

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Computing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats

Cited by 9 publications

References 5 publications

The G-invariant and catenary data of a matroid

The G-invariant and catenary data of a matroid

On Binary Matroid Minors and Applications to Data Storage over Small Fields

Matroids with different configurations and the same $\mathcal{G}$-invariant

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