Generalized Laminar Matroids

TL;DR

This paper introduces and explores generalized classes of laminar matroids, providing structural properties and characterizations, including excluded-minor characterizations for 2-laminar and 2-closure-laminar matroids.

Contribution

It defines the classes of k-closure-laminar and k-laminar matroids, analyzes their properties, and characterizes the 2-laminar and 2-closure-laminar classes via excluded minors.

Findings

2-laminar and 2-closure-laminar classes are minor-closed.

Structural properties of k-laminar classes are established.

Excluded-minor characterizations for 2-laminar and 2-closure-laminar classes are provided.

Abstract

Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats form a chain under inclusion; $M$ is laminar if and only if, for every $1$-element independent set $X$, the Hamiltonian flats of $M$ containing $X$ form a chain under inclusion. We generalize these notions to define the classes of $k$-closure-laminar and $k$-laminar matroids. This paper focuses on structural properties of these classes noting that, while the second class is always minor-closed, the first is if and only if $k \le 3$. The main results are excluded-minor characterizations for the classes of 2-laminar and 2-closure-laminar matroids.