Matroid, Ideal, Ultrafilter, Tangle, and so on: Reconsideration of Obstruction to linear decomposition

TL;DR

This paper investigates the challenges to linear decomposition in connectivity systems, emphasizing the role of matroids, filters, and related structures to better understand width parameters like linear branch width.

Contribution

It offers a new perspective on obstructions to linear decomposition by analyzing matroids, filters, and connectivity systems, broadening the understanding of width parameters.

Findings

Identifies key obstructions to linear decomposition.

Highlights the relevance of matroids and filters in connectivity.

Provides insights into the structure of width parameters.

Abstract

The investigation of width parameters in both graph and algebraic contexts has attracted considerable interest. Among these parameters, the linear branch width has emerged as a crucial measure. In this concise paper, we explore the concept of linear decomposition, specifically focusing on the single filter in a connectivity system. Additionally, we examine the relevance of matroids, antimatroids, and greedoids in the context of connectivity systems. Our primary objective in this study is to shed light on the impediments to linear decomposition from multiple perspectives.