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≤3. We give excluded-minor characterizations of the classes of 2-laminar and 2-closure-laminar matroids
