On the relative importance of excluded minors

TL;DR

This paper characterizes superfluous subsets within collections of excluded minors in matroid theory, identifying when removing certain minors results in only finitely many 3-connected matroids being excluded.

Contribution

It provides a complete characterization of superfluous subsets of six well-known collections of excluded minors in matroid theory.

Findings

Identifies superfluous subsets in six key collections of excluded minors

Shows conditions under which minor sets are superfluous

Provides a framework for simplifying excluded minor sets

Abstract

If EE is a set of matroids, then ex(EE) denotes the set of matroids that have no minor isomorphic to a member of EE. If EE' is a subset of EE, we say that EE' is /superfluous/ if ex(EE - EE') - ex(EE) contains only finitely many 3-connected matroids. We characterize the superfluous subsets of six well-known collections of excluded minors.