$N$-detachable pairs in 3-connected matroids II: life in $X$

TL;DR

This paper investigates the structural properties of 3-connected matroids lacking N-detachable pairs, identifying specific separators and configurations that influence the existence of such pairs.

Contribution

It extends previous work by characterizing matroids with structured sets X, revealing conditions under which N-detachable pairs must exist or specific separators occur.

Findings

Identifies five particular 3-separators in matroids without N-detachable pairs.

Shows structured set X leads to either an N-detachable pair or specific separators.

Provides a detailed structural analysis of 3-connected matroids in this context.

Abstract

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the second in a series of three papers where we describe the structures that arise when it is not possible to find an $N$-detachable pair in $M$. In the first paper in the series, we showed that, under mild assumptions, either $M$ has an $N$-detachable pair, $M$ has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs, or there is a 3-separating set $X$ with certain strong structural properties. In this paper, we analyse matroids with such a structured set $X$, and prove that they have either an $N$-detachable pair, or one of five particular 3-separators that can appear in a matroid with no $N$-detachable pairs.