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Matroid Combinatorics Graphic matroid Mathematics Matroid partitioning Rank (graph theory) Bounded function Oriented matroid Discrete mathematics Set (abstract data type) Weighted matroid Computer science Mathematical analysis Programming language

Ragnar Freij-Hollanti , Relinde Jurrius , O. I. Kuznetsova ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.37236/11327 · OA: W4362699102

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 $\lvert M\rvert-r(M)$ and that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.