Laminar Matroids

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.Comment: 17 page

Generalized Laminar Matroids

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.Comment: 12 page

Lattice path matroids: structural properties

This paper studies structural aspects of lattice path matroids. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.Postprint (published version

On Selected Subclasses of Matroids

Matroids were introduced by Whitney to provide an abstract notion of independence. In this work, after giving a brief survey of matroid theory, we describe structural results for various classes of matroids. A connected matroid $M$ is unbreakable if, for each of its flats $F$, the matroid $M/F$ is connected%or, equivalently, if $M^*$ has no two skew circuits. . Pfeil showed that a simple graphic matroid $M(G)$ is unbreakable exactly when $G$ is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids. A laminar family is a collection \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 \A, let \I be %the set \{I:|I\cap A| \leq c(A)\text{ for all A\in\A}\}. Then \I is the collection of independent sets of a (laminar) matroid on $E$. We characterize the class of laminar matroids by their excluded minors and present a way to construct all laminar matroids using basic operations. %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 its Hamiltonian flats form a chain under inclusion; $M$ is laminar 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. The second class is always minor-closed, and the first is if and only if $k \le 3$. We give excluded-minor characterizations of the classes of 2-laminar and 2-closure-laminar matroids