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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 973920, 27 pages
http://dx.doi.org/10.1155/2012/973920
Research Article

Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

1School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
2School of Computer Science and Engineering, XinJiang University of Finance and Economics, Urumqi 830012, China
3Lab of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, China

Received 4 February 2012; Revised 30 April 2012; Accepted 18 May 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 Jianguo Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. Some new properties, which have not been found in rough sets, are obtained. Furthermore, by defining the concept of lower approximation number, the rank function of some subset of the universe and the approximations of the subset are connected. Finally, the relationships between the two types of matroids are discussed, and the result shows that they are just dual matroids.