Laminar Matroids

TL;DR

This paper studies laminar matroids, a special class of matroids defined by laminar families, and introduces methods for their compact representation, characterization via excluded minors, and construction from basic operations.

Contribution

It provides a compacting method for laminar presentations, characterizes laminar matroids by excluded minors, and offers a construction framework for all laminar matroids.

Findings

Characterization of laminar matroids by excluded minors

A method for compacting laminar presentations

Construction of all laminar matroids from basic operations

Abstract

A laminar family is a collection $\mathscr{A}$ of subsets of a set $E$ such that, for any two intersecting sets, one is contained in the other. For a capacity function $c$ on $\mathscr{A}$, let $\mathscr{I}$ be $\{I:|I\cap A| \leq c(A)\text{ for all $A\in\mathscr{A}$}\}$. Then $\mathscr{I}$ is the collection of independent sets of a (laminar) matroid on $E$. We present a method of compacting laminar presentations, characterize the class of laminar matroids by their excluded minors, present a way to construct all laminar matroids using basic operations, and compare the class of laminar matroids to other well-known classes of matroids.