In this paper, we generalize three types of rough set models based on j -neighborhood space (i.e, type 1 j -neighborhood rough set, type 2 j -neighborhood rough set, and type 3 j -neighborhood rough set), and investigate some of their basic properties. Also, we present another three types of rough set models based on j -adhesion neighborhood space (i.e, type 4 j -adhesion neighborhood rough set, type 5 j -adhesion neighborhood rough set, and type 6 j -adhesion neighborhood rough set). The fundamental properties of approximation operators based on j -adhesion neighborhood space are established. The relationship between the properties of these types is explained. Finally, according to j -adhesion neighborhood space, we give a comparison between the Yao’s approach and our approach.

Neighborhood systems are used to approximate graphs as finite topological structures. Throughout this article, we construct new types of eight neighborhoods for vertices of an arbitrary graph, say, j-adhesion neighborhoods. Both notions of Allam et al. and Yao will be extended via j-adhesion neighborhoods. We investigate new types of j-lower approximations and j-upper approximations for any subgraph of a given graph. Then, the accuracy of these approximations will be calculated. Moreover, a comparison between accuracy measures and boundary regions for different kinds of approximations will be discussed. To generate j-adhesion neighborhoods and rough sets on graphs, some algorithms will be introduced. Finally, a sample of a chemical example for Walczak will be introduced to illustrate our proposed methods.

Covering-based rough sets are important generalizations of the classical rough sets of Pawlak. In this paper, by means of j -neighborhoods, complementary j -neighborhoods and j -adhesions, we build some new different types of j -covering approximations based rough sets and study related properties. Also, we explore the relationships between the considered j -covering approximations and investigate the properties of them. Using different neighborhoods, some different general topologies are generated as topologies induced from a binary relation. Finally, an interesting application of the new types of covering-based rough sets to the rheumatic fever is given.