Vertically N-contractible elements in 3-connected matroids

TL;DR

This paper generalizes the concept of vertically N-contractible elements in 3-connected matroids, extending previous binary case results to non-binary cases and exploring related properties like 3-roundedness.

Contribution

It extends Costalonga's results on vertically N-contractible elements from binary to non-binary matroids and applies these findings to properties akin to 3-roundedness.

Findings

Generalization of Costalonga's results to non-binary matroids

Characterization of obstructions for 4-independent sets of contractible elements

Application to properties similar to 3-roundedness

Abstract

In this paper we establish a variation of the Splitter Theorem. Let $M$ and $N$ be simple 3-connected matroids. We say that $x\in E(M)$ is vertically $N$-contractible if $si(M/x)$ is a 3-connected matroid with an $N$-minor. Whittle (for $k=1,2$) and Costalonga(for $k=3$) proved that, if $r(M)- r(N)\ge k$, then $M$ has a $k$-independent set $I$ of vertically $N$-contractible elements. Costalonga also characterized an obstruction for the existence of such a 4-independent set $I$ in the binary case, provided $r(M)-r(N)\ge 5$, and improved this result when $r(M)-r(N)\ge 6$, and in the graphic case. In this paper we generalize the results of Costalonga to the non-binary case. Moreover, we apply our results to the study of properties similar to 3-roundedness in classes of matroids.