Templates for Binary Matroids

TL;DR

This paper introduces binary frame templates for constructing binary matroids, classifies minimal nontrivial templates, and applies these to determine growth rates and properties of certain minor-closed classes of binary matroids.

Contribution

It defines a preorder on binary frame templates, classifies minimal nontrivial templates, and applies these results to analyze growth rates and connectivity properties of binary matroids.

Findings

Identifies minimal nontrivial binary frame templates.

Determines growth rates of specific minor-closed classes.

Characterizes highly-connected 1-flowing matroids.

Abstract

A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial templates that are minimal with respect to this order. As an application of our main result, we determine the eventual growth rates of certain minor-closed classes of binary matroids, including the class of binary matroids with no minor isomorphic to PG(3,2). Our main result applies to all highly-connected matroids in a class, not just those of maximum size. As a second application, we characterize the highly-connected 1-flowing matroids.