基于粗糙集的拟阵连通性
Connectivity for matroids based on rough sets
摘要：
在数学和计算机科学中，连通性是拟阵理论的基本概念之一：它要求最小数量的元素需要被删除，以断开其余节点彼此。它与网络流问题的理论密切相关。拟阵的连通性是衡量其网络鲁棒性的重要指标。因此，研究拟阵连通的条件是十分必要的。本文利用基于关系的粗糙集研究拟阵的连通性。首先，从一般拟阵出发，引入了一种对称传递关系，并从拟阵的角度研究了它的性质。此外，通过一般拟阵引入的关系，将无向图进行了推广。具体地说，图的连通性可以通过基于关系的粗糙集来研究。其次，利用基于关系的粗糙集理论研究了拟阵的连通性，给出了一般拟阵连通的条件。最后，证明了具有某些特殊性质的一般拟阵与其导出无向图的连通性是等价的。这些结果显示了基于关系的粗糙集的拟阵的一个重要应用。
英文摘要：
In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems. The connectivity of a matroid is an important measure of its robustness as a network. Therefore, it is very necessary to investigate the conditions under which a matroid is connected. In this paper, the connectivity for matroids is studied through relation-based rough sets. First, a symmetric and transitive relation is introduced from a general matroid and its properties are explored from the viewpoint of matroids. Moreover, through the relation introduced by a general matroid, an undirected graph is generalized. Specifically, the connection of the graph can be investigated by the relation-based rough sets. Second, we study the connectivity for matroids by means of relation-based rough sets and some conditions under which a general matroid is connected are presented. Finally, it is easy to prove that the connectivity for a general matroid with some special properties and its induced undirected graph is equivalent. These results show an important application of relation-based rough sets to matroids.
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