Topological Structure of Generalized Rough Graphs

Samah Sarmad,Yousif Yaqoub Yousif

doi:10.30526/34.1.2559

Abstract

The main purpose of this paper, is to introduce a topological space , which is induced by reflexive graph and tolerance graph , such that may be infinite. Furthermore, we offered some properties of such as connectedness, compactness, LindelÃ¶f and separate properties. We also study the concept of approximation spaces and get the sufficient and necessary condition that topological space is approximation spaces.

Highlights

Graph theory [1] is a tool for optimization and solving practical application in all fields such as engineering study and representation of economic and social networks, complex general systems, information theory and others
Rough set was offered by Pawlak [2] as a method for dealing with uncertainly of imprecise data, the equivalence relation is the cornerstone of Pawlak,s theory of rough set
We will study through this part, the properties of the topological space (D, τD), where (D, τD) is produced by a reflexive graph D

Summary

Introduction

Graph theory [1] is a tool for optimization and solving practical application in all fields such as engineering study and representation of economic and social networks, complex general systems, information theory and others. Z. Li [3] offered the concept of transmitting expression of relation and produced several important results of rough sets topological properties. IfDis a reflexive graph, (D, τD) is named the topological space produced by D. (3) Let (ɍ, u) ∈ E(Dβ), we have to show that (ɍ, u) ∈ E(Dα).Since Dβ the transmitting expression of Dα, (ɍ, u) ∈ E(Dβ)if and only if (ɍ, u) ∈ Dα or there exists {v1, v2, v3, ... (1) Let (D, τD) be a topological space and BD a base of (D, τD), where (D, τD) is induced by a reflexive graph D. Let D = (V(D), E(D)) be a nonempty graph, d: V(D) × V(D) ⟶ [0, +∞) is called pseudo-metric map on D, if for all ɍ, v, u ∈ V(D),. D is named a pseudo-discrete space if Q ⊆ D is open in D if and only if Q is closed in D

The Properties of Topological Spaces Induced by a Reflexive Graph

The Properties of Topological Spaces generated by a Tolerance Graph

Approximation spaces on digraph

3.Conclusion