TL;DR
This paper characterizes threshold matroids through their bases, provides an example of a shifted matroid that is not threshold, and offers algorithms and formulas related to their enumeration and recognition.
Contribution
It introduces a structural characterization of threshold matroids, distinguishes shifted matroids that are not threshold, and develops algorithms and formulas for their enumeration and recognition.
Findings
Almost all shifted matroids are not threshold.
A polynomial-time algorithm for checking if a matroid is threshold.
A formula for counting isomorphism classes of threshold matroids.
Abstract
We characterize the class of threshold matroids by the structure of their defining bases. We also give an example of a shifted matroid which is not threshold, answering a question of Deza and Onn. We conclude by exploring consequences of our characterization of threshold matroids: We give a formula for the number of isomorphism classes of threshold matroids on a ground set of size n. This enumeration shows that almost all shifted matroids are not threshold. We also present a polynomial-time algorithm to check if a matroid is threshold and provide alternative and simplified proofs of some of the main results of Deza and Onn.
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Taxonomy
TopicsPhotonic Crystals and Applications · Plasmonic and Surface Plasmon Research