$N$-detachable pairs in 3-connected matroids I: unveiling $X$

TL;DR

This paper investigates the structural properties of 3-connected matroids lacking N-detachable pairs, identifying specific 3-separators or sets with strong structural features, advancing understanding of matroid connectivity and minors.

Contribution

It characterizes the structure of 3-connected matroids without N-detachable pairs, introducing new types of 3-separators and structural conditions.

Findings

Existence of a 3-separating set with strong properties

Identification of three particular 3-separators

Framework for analyzing matroids without N-detachable pairs

Abstract

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. We say that 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 first 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 this paper, we prove that if $M$ has no $N$-detachable pairs, then either $M$ has a 3-separating set, which we call $X$, with certain strong structural properties, or $M$ has one of three particular 3-separators that can appear in a matroid with no $N$-detachable pairs.