Combinatorial Derived Matroids

TL;DR

This paper introduces a new combinatorial definition of derived matroids applicable to any matroid, analyzes their properties, and provides examples including uniform, Vamos, and graphical matroids.

Contribution

It presents a novel, fully combinatorial definition of derived matroids for arbitrary matroids, extending previous approaches.

Findings

The rank of the derived matroid is bounded by |M|-r(M).

Derived matroids are connected if and only if the original matroid is connected.

Examples include derived matroids of uniform, Vamos, and graphical matroids.

Abstract

Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $\delta M$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $\delta M$ is bounded from above by $|M|-r(M)$, that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the V\'amos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.