Triangle-roundedness in matroids

TL;DR

This paper investigates the property of triangle-roundedness in matroids, extending previous results to non-binary cases and demonstrating that the matroid of the complete graph on five vertices is triangle-rounded within regular matroids.

Contribution

It generalizes Reid's result by removing a key condition and extends the concept of triangle-roundedness to non-binary matroids, with an application to regular matroids.

Findings

Extended Reid's result to non-binary matroids

Proved $M(K_5)$ is triangle-rounded in regular matroids

Dropped the element existence condition in the main theorem

Abstract

A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We strengthen this result by dropping the condition that such element $e$ exists and proving that there is a $3$-connected minor $M'$ of $M$ with an $N$-minor $N'$ such that $T$ is a triangle of $M'$ and $E(M')-E(N')\subseteq T$. This result is extended to the non-binary case and, as an application, we prove that $M(K_5)$ is triangle-rounded in the class of the regular matroids.