在拟阵和图论领域，如何刻画一个对子式运算封闭的拟阵类或者图类，一直是个重要的研究领域。在该领域，产生了许多经典的、重要的、基础性的成果。如平面图的排除子式、图子式理论（Lovasz称之为离散数学中里程碑似的成果）和著名的Rota猜想等。J.Geelen等人猜测，有限域F可表示的任意一个拟阵都有一个树分解，使得它的每个部分大体上是可表示框架拟阵、可表示框架拟阵的对偶拟阵、或者F的某个真子域可表示拟阵。因为框架拟阵（或者提升图拟阵）不仅有无限多个排除子式，而且对任意的多项式p(.)，知道p(|E(M)|)个E(M)的子集的秩，都不能判断出一个拟阵M是不是框架拟阵或者提升图拟阵。因此，如何刻画可表示框架拟阵和可表示提升图拟阵是大家关注的一个问题。本项目的主要目的是研究这两类拟阵和bicircular拟阵的刻画问题，争取解决该领域的几个有影响力的猜想。

How to characterize a minor-closed class X is always an important research field in graph theory and matroid theory, especially when X is important. There are many important and fundamental results in this field. For example, excluded minors for the class of planar graphs, the Graph Minor Project, and Rota’s Conjecture which is one of the most famous conjecture in matroid theory. Geelen, Gerards and Whittle conjectured that for any proper minor-closed class of F-representable matroids, each matroid in this class admits a tree-like decomposition such that each part is either essentially a representable frame matroid, or is essentially the dual of a representable frame matroid, or is essentially represented over a proper subfield of F. However, the class of frame matroids and the class of lifted-graphic matroids have infinitely many excluded minors; and there is no polynomial function p(.) with the property that a matroid M can be determined to be either a lifted-graphic or frame matroid using at most p(|E(M)|) rank evaluations. Hence, characterizing the class of representable frame matroids and the class of representable lifted-graphic matroids becomes important. The main purpose of this project is to characterize the two classes of matroids and the class of bicircular matroids.