Contracting an element from a cocircuit

TL;DR

This paper investigates conditions under which contracting an element from a cocircuit in a 3-connected matroid preserves 3-connectivity and an N-minor, providing structural insights into matroid minors.

Contribution

It establishes new structural results on contracting elements from cocircuits in 3-connected matroids while maintaining minors and connectivity.

Findings

Existence of an element x in C* with preserved 3-connectivity and N-minor after contraction.

Presence of a four-element fan containing two elements of C* and an element x with similar properties.

Structural characterization of matroids with cocircuit contraction preserving minors.

Abstract

We consider the situation that M and N are 3-connected matroids such that |E(N)| > 3 and C* is a cocircuit of M with the property that M/y has an N-minor for some y in C*. We show that either there is an element x in C* such that si(M/x) or co(si(M/x)) is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of C* and an element x such that si(M/x) is 3-connected with an N-minor.