Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# On the Structure of Triangle-Free 3-Connected Matroids

Version 1 : Received: 15 February 2021 / Approved: 17 February 2021 / Online: 17 February 2021 (09:42:30 CET)

How to cite: dos Santos, J.C. On the Structure of Triangle-Free 3-Connected Matroids. Preprints 2021, 2021020356 (doi: 10.20944/preprints202102.0356.v1). dos Santos, J.C. On the Structure of Triangle-Free 3-Connected Matroids. Preprints 2021, 2021020356 (doi: 10.20944/preprints202102.0356.v1).

## Abstract

Let $\mathcal{N}$ be an arbitrary class of matroids, closed under isomorphism. For $k$ a positive integer, we say that $M \in \mathcal{N}$ is \emph{$k$-minor-irreducible} if $M$ has no minor $N \in \mathcal{N}$ such that $1\leq\left|E\left(M\right)\right|-\left|E\left(N\right)\right|\leq k$. Tutte's Wheels and Whirls Theorem establish that, up to isomorphism, there are only two families of 1-minor-irreducible matroids in the class of 3-connected matroids. More recently, Lemos classified the 3-minor-irreducibles with at least 14 elements in the class of triangle-free 3-connected matroids. Here we prove a local characterization for the 2-minor-irreducible matroids with at least 11 elements in the class of triangle-free 3-connected matroids. This local characterization is used to establish two new families of 2-minor-irreducible matroids in this class.

## Keywords

Matroid; 3-connected; Minor; Irreducible; Triangle; Triad

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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