Non-Separating Cocircuits and Graphicness in Matroids

TL;DR

This paper explores the relationship between non-separating cocircuits and the graphicness of 3-connected binary matroids, establishing bounds on elements avoiding certain cocircuits and proposing a conjecture related to regular matroids with specific minors.

Contribution

It generalizes Lemos's characterization of non-graphic matroids by analyzing the size of Y(M) in relation to minors and regularity, and introduces a conjecture on the structure of regular matroids with M(K_{3,3})-minors.

Findings

Y(M) is very large for non-graphic matroids without M(K_{3,3})-minors

In such cases, |E(M)-Y(M)| ≤ 1

Conjecture: For regular matroids with M(K_{3,3})-minors, r_M(E(M)-Y(M)) ≤ 2

Abstract

Let $M$ be a 3-connected binary matroid and let $Y(M)$ be the set of elements of $M$ avoiding at least $r(M)+1$ non-separating cocircuits of $M$. Lemos proved that $M$ is non-graphic if and only if $Y(M)\neq\emp$. We generalize this result when by establishing that $Y(M)$ is very large when $M$ is non-graphic and $M$ has no $M\s(K_{3,3}"')$-minor if $M$ is regular. More precisely that $|E(M)-Y(M)|\le 1$ in this case. We conjecture that when $M$ is a regular matroid with an $M\s(K_{3,3})$-minor, then $r\s_M(E(M)-Y(M))\le 2$. The proof of such conjecture is reduced to a computational verification.