Table of Contents
ISRN Applied Mathematics
VolumeΒ 2012, Article IDΒ 240315, 23 pages
http://dx.doi.org/10.5402/2012/240315
Research Article

Near Approximations in πΊπ‘š-Closure Spaces

1Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
2Department of Physics and Mathematics, Faculty of Engineering, Tanta University, Tanta 31111, Egypt
3Department of Mathematics, Faculty of Education Ibn-Al-Haitham, Baghdad University, Baghdad 4150, Iraq

Received 21 December 2011; Accepted 8 February 2012

Academic Editors: I.Β Doltsinis and H.-T.Β Hu

Copyright Β© 2012 M. E. Abd El-Monsef 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

Most real-life situations need some sort of approximation to fit mathematical models. The beauty of using topology in approximation is achieved via obtaining approximation for qualitative subgraphs without coding or using assumption. The aim of this paper is to apply near concepts in the πΊπ‘š-closure approximation spaces. The basic notions of near approximations are introduced and sufficiently illustrated. Near approximations are considered as mathematical tools to modify the approximations of graphs. Moreover, proved results, examples, and counterexamples are provided.

1. Introduction

The theory of rough sets, proposed by Pawlak [1], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of lower and upper approximation in rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules. The notions of closure operator and closure system are very useful tools in several sections of mathematics, as an example, in algebra [2–4], topology [5–7], and computer science theory [8, 9]. Many works have appeared recently, for example, in structural analysis [10, 11], in chemistry [12], and in physics [13]. The purpose of the present work is to put a starting point for the application of abstract topological graph theory in the rough set analysis. Also, we will integrate some ideas in terms of concept in topological graph theory. Topological graph theory is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics but also in many real-life applications. We believe that topological graph structure will be an important base for modification of knowledge extraction and processing.

2. Preliminaries

This section presents a review of some fundamental notions of Pawlak’s rough sets [1, 14, 15] and πΊπ‘š-closure spaces [10, 11].

2.1. Fundamental Notions of Uncertainty

Motivation for rough set theory has come from the need to represent subsets of a universe in terms of equivalence classes of a partition of that universe. The partition characterizes a topological space, called approximation space 𝐾=(𝑋,𝑅), where 𝑋 is a set called the universe and 𝑅 is an equivalence relation [15, 16]. The equivalence classes of 𝑅 are also known as the granules, elementary sets or blocks; we will use 𝑅π‘₯βŠ†π‘‹ to denote the equivalence class containing π‘₯βˆˆπ‘‹. In the approximation space, we consider two operators, the upper and lower approximations of subsets: let π΄βŠ†π‘‹, then the lower approximation (resp., the upper approximation) of 𝐴 is given by𝐿(𝐴)=π‘₯βˆˆπ‘‹βˆΆπ‘…π‘₯βŠ†π΄ξ€Ύξ€·respξ€½.,π‘ˆ(𝐴)=π‘₯βˆˆπ‘‹βˆΆπ‘…π‘₯.βˆ©π΄β‰ πœ™ξ€Ύξ€Έ(2.1) Boundary, positive, and negative regions are also defined:Bd𝑅(𝐴)=π‘ˆ(𝐴)βˆ’πΏ(𝐴),POS𝑅(𝐴)=𝐿(𝐴),NEG𝑅(𝐴)=π‘‹βˆ’π‘ˆ(𝐴).(2.2)

In an approximation space 𝐾=(𝑋,𝑅), if 𝐴 and 𝐡 are two subsets of 𝑋, then directly from the definitions of lower and upper approximations, we can get the following properties of the lower and upper approximations [15]:(1)𝐿(𝐴)βŠ†π΄βŠ†π‘ˆ(𝐴), (2)𝐿(πœ™)=π‘ˆ(πœ™)=πœ™and𝐿(𝑋)=π‘ˆ(𝑋)=𝑋, (3)π‘ˆ(𝐴βˆͺ𝐡)=π‘ˆ(𝐴)βˆͺπ‘ˆ(𝐡), (4)𝐿(𝐴∩𝐡)=𝐿(𝐴)∩𝐿(𝐡), (5)Ifπ΄βŠ†π΅,then𝐿(𝐴)βŠ†πΏ(𝐡), (6)Ifπ΄βŠ†π΅,thenπ‘ˆ(𝐴)βŠ†π‘ˆ(𝐡), (7)𝐿(𝐴βˆͺ𝐡)βŠ‡πΏ(𝐴)βˆͺ𝐿(𝐡), (8)π‘ˆ(𝐴∩𝐡)βŠ†π‘ˆ(𝐴)βˆ©π‘ˆ(𝐡), (9)𝐿(𝐴𝑐)=[π‘ˆ(𝐴)]𝑐, (10)π‘ˆ(𝐴𝑐)=[𝐿(𝐴)]𝑐, (11)𝐿(𝐿(𝐴))=π‘ˆ(𝐿(𝐴))=𝐿(𝐴), (12)π‘ˆ(π‘ˆ(𝐴))=𝐿(π‘ˆ(𝐴))=π‘ˆ(𝐴).

The inexactness of a set is due to the existence of a boundary region. The greater of the boundary region of a set, means the Pawlak [1], introduced the accuracy measure which is considered as a numerical characterization of imprecision. The following definition gives the accuracy measure of a subset π΄βŠ†π‘‹ in approximation space 𝐾=(𝑋,𝑅).

Definition 2.1. Let 𝐾=(𝑋,𝑅) be an approximation space. The accuracy measure of a subset π΄βŠ†π‘‹ is defined by πœ‚(𝐴) and define by ||||πœ‚(𝐴)=𝐿(𝐴)||||,π‘ˆ(𝐴)where||||π‘ˆ(𝐴)β‰ 0.(2.3)

The accuracy measure is also called the accuracy of approximation.

2.2. Fundamental Notions of πΊπ‘š-Closure Spaces

In this section, we introduce the concepts of closure operators on digraphs; several known topological properties on the obtained πΊπ‘š-closure spaces are studied.

Definition 2.2 (see [10, 11]). Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a digraph, 𝑃(𝑉(𝐺)) its power set of all subgraphs of 𝐺, and ClπΊβˆΆπ‘ƒ(𝑉(𝐺))→𝑃(𝑉(𝐺)) a mapping associating with each subgraph 𝐻=(𝑉(𝐻),𝐸(𝐻)); a subgraph Cl𝐺(𝑉(𝐻))βŠ†π‘‰(𝐺) is called the closure subgraph of 𝐻 such that Cl𝐺(𝑉(𝐻))=𝑉(𝐻)βˆͺπ‘£βˆˆπ‘‰(𝐺)βˆ’π‘‰(𝐻);hv.∈𝐸(𝐺)βˆ€β„Žβˆˆπ‘‰(𝐻)(2.4) The operation Cl𝐺 is called graph closure operator, and the pair (𝐺,ℱ𝐺) is called 𝐺-closure space, where ℱ𝐺 is the family of elements of Cl𝐺. Evidently Cl𝐺(𝑉(𝐻))=∩{𝑉(𝐹);𝑉(𝐹)βˆˆβ„±πΊand𝑉(𝐻)βŠ†π‘‰(𝐹)}. The dual of the graph closure operator Cl𝐺 is the graph interior operator IntπΊβˆΆπ‘ƒ(𝑉(𝐺))→𝑃(𝑉(𝐺)) defined by Int𝐺(𝑉(𝐻))=𝑉(𝐺)βˆ’Cl𝐺(𝑉(𝐺)βˆ’π‘‰(𝐻)) for all subgraph π»βŠ†πΊ. A family of elements of Int𝐺 is called interior subgraph of 𝐻 and denoted by 𝒯𝐺. It is clear that (𝐺,𝒯𝐺) is a topological space. Evidently Int𝐺(𝑉(𝐻))=βˆͺ{𝑉(O);𝑉(O)βˆˆπ’―πΊand𝑉(O)βŠ†π‘‰(𝐻)}. Then the domain of Cl𝐺 is equal to the domain of Int𝐺 and also Cl𝐺(𝑉(𝐻))=𝑉(𝐺)βˆ’Int𝐺(𝑉(𝐺)βˆ’π‘‰(𝐻)). A subgraph 𝐻 of 𝐺-closure space (𝐺,𝒯𝐺) is called closed subgraph if Cl𝐺(𝑉(𝐻))=𝑉(𝐻). It is called open subgraph if its complement is closed subgraph, that is, Cl𝐺(𝑉(𝐺)βˆ’π‘‰(𝐻))=𝑉(𝐺)βˆ’π‘‰(𝐻), or equivalently Int𝐺(𝑉(𝐻))=𝑉(𝐻).

Example 2.3. Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a digraph such that:𝑉(𝐺)={𝑣1,𝑣2,𝑣3,𝑣4}, 𝐸(𝐺)={(𝑣1,𝑣2),(𝑣1,𝑣3),(𝑣2,𝑣1),(𝑣2,𝑣3),(𝑣4,𝑣3)}, for more details (Table 1)240315.fig.001ℱ𝐺={𝑉(𝐺),πœ™,{𝑣3},{𝑣3,𝑣4},{𝑣1,𝑣2,𝑣3}}, 𝒯𝐺={𝑉(𝐺),πœ™,{𝑣4},{𝑣1,𝑣2},{𝑣1,𝑣2,𝑣4}}.
We obtain a new definition to construct topological closure spaces from 𝐺-closure spaces by redefining graph closure operator on the resultant subgraphs as a domain of the graph closure operator and stop when the operator transfers each subgraph to itself.

tab1
Table 1

Definition 2.4 (see [10, 11]). Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a digraph and ClπΊπ‘šβˆΆπ‘ƒ(𝑉(𝐺))→𝑃(𝑉(𝐺)) an operator such that:(a)It is called πΊπ‘š-closure operator if ClπΊπ‘š(𝑉(𝐻))=Cl𝐺(Cl𝐺(…Cl𝐺(𝑉(𝐻)))), π‘š-times, for every subgraph π»βŠ†πΊ,(b)it is called πΊπ‘š-topological closure operator if ClπΊπ‘š+1(𝑉(𝐻))=ClπΊπ‘š(𝑉(𝐻)) for all subgraph π»βŠ†πΊ.The space (𝐺,β„±πΊπ‘š) is called πΊπ‘š-closure space.

Example 2.5. Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a digraph such that:𝑉(𝐺)={𝑣1,𝑣2,𝑣3,𝑣4}, 𝐸(𝐺)={(𝑣1,𝑣3),(𝑣2,𝑣1),(𝑣2,𝑣3),(𝑣3,𝑣4),(𝑣4,𝑣1)},for more details (see Table 2)240315.fig.002ℱ𝐺2={𝑉(𝐺),πœ™,{𝑣1,𝑣3,𝑣4}}, 𝒯𝐺2={𝑉(𝐺),πœ™,{𝑣2}}.

tab2
Table 2

Proposition 2.6 (see [10]). Let (𝐺,β„±πΊπ‘š) be a πΊπ‘š-closure space. If 𝐻 and 𝐾 are two subgraphs of 𝐺 such that π»βŠ†πΎβŠ†πΊ, then

ClπΊπ‘š(𝑉(𝐻))βŠ†ClπΊπ‘š(𝑉(𝐾)),IntπΊπ‘š(𝑉(𝐻))βŠ†IntπΊπ‘š(𝑉(𝐾)).(2.5)

Proposition 2.7 (see [10]). Let (𝐺,β„±πΊπ‘š) be a πΊπ‘š-closure space. If 𝐻 and 𝐾 are two subgraphs of 𝐺, then(a)ClπΊπ‘š(𝑉(𝐻)βˆͺ𝑉(𝐾))=ClπΊπ‘š(𝑉(𝐻))βˆͺClπΊπ‘š(𝑉(𝐾)), (b)IntπΊπ‘š(𝑉(𝐻)βˆ©π‘‰(𝐾))=IntπΊπ‘š(𝑉(𝐻))∩IntπΊπ‘š(𝑉(𝐾)).

Proposition 2.8 (see [10]). Let (𝐺, β„±πΊπ‘š) be a πΊπ‘š-closure space. If 𝐻 and 𝐾 are two subgraphs of 𝐺, then(a)ClπΊπ‘š(𝑉(𝐻)βˆ©π‘‰(𝐾))βŠ†ClπΊπ‘š(𝑉(𝐻))∩ClπΊπ‘š(𝑉(𝐾)), and(b)IntπΊπ‘š(𝑉(𝐻))βˆͺIntπΊπ‘š(𝑉(𝐾))βŠ†IntπΊπ‘š(𝑉(𝐻)βˆͺ𝑉(𝐾)).

Remark 2.9. The converse of Proposition 2.8 need not be true in general, as the following example (Example  2.3 in [10]).

Definition 2.10 (see [10]). Let (𝐺,β„±πΊπ‘š) be a πΊπ‘š-closure space and π»βŠ†πΊ; the boundary of 𝐻 is denoted by BdπΊπ‘š(𝑉(𝐻)) and is defined by BdπΊπ‘š(𝑉(𝐻))=ClπΊπ‘š(𝑉(𝐻))βˆ’IntπΊπ‘š(𝑉(𝐻)).(2.6)

Proposition 2.11 (see [10]). Let (𝐺,β„±πΊπ‘š) be a πΊπ‘š-closure space and π»βŠ†πΊ, then(a)BdπΊπ‘š(𝑉(𝐻))=ClπΊπ‘š(𝑉(𝐻))∩ClπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻)), (b)BdπΊπ‘š(𝑉(𝐻))=BdπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻)), (c)ClπΊπ‘š(𝑉(𝐻))=𝑉(𝐻)βˆͺBdπΊπ‘š(𝑉(𝐻)), (d)IntπΊπ‘š(𝑉(𝐻))=𝑉(𝐻)βˆ’BdπΊπ‘š(𝑉(𝐻)).

By a similar way of definitions of regular open set [17], semiopen set [18], preopen set [19], Ξ³-open set [20] (b-open set [21]), Ξ±-open set [22], and Ξ²-open set [23] (=semi-pre-open set [24]), we introduce the following definitions which are essential for our present study. In πΊπ‘š-closure space (𝐺, β„±πΊπ‘š) the subgraph 𝐻 of 𝐺 is called(a)regular open subgraph [10] (briefly 𝑅-osg) if 𝑉(𝐻)=IntπΊπ‘š(ClπΊπ‘š(𝑉(𝐻))),(b) semiopen subgraph [10] (briefly 𝑆-osg) if 𝑉(𝐻)βŠ†ClπΊπ‘š(IntπΊπ‘š(𝑉(𝐻))),(c) preopen subgraph [10] (briefly 𝑃-osg) if 𝑉(𝐻)βŠ†IntπΊπ‘š(ClπΊπ‘š(𝑉(𝐻))),(d)Ξ³-open subgraph (briefly Ξ³-osg) if 𝑉(𝐻)βŠ†ClπΊπ‘š(IntπΊπ‘š(𝑉(𝐻)))βˆͺIntπΊπ‘š(ClπΊπ‘š(𝑉(𝐻))),(e)Ξ±-open subgraph [10] (briefly Ξ±-osg) if 𝑉(𝐻)βŠ†IntπΊπ‘š(ClπΊπ‘š(IntπΊπ‘š(𝑉(𝐻))),(f)Ξ²-open subgraph [10] (briefly Ξ²-osg) if 𝑉(𝐻)βŠ†ClπΊπ‘š(IntπΊπ‘š(ClπΊπ‘šπ‘‰(𝐻))).

The complement of an 𝑅-osg (resp., 𝑆-osg, 𝑃-osg, Ξ³-osg, Ξ±-osg, and Ξ²-osg) is called 𝑅-closed subgraph (briefly 𝑅-csg) (resp., S-csg, 𝑃-csg, Ξ³-csg, Ξ±-csg, and Ξ²-csg).

The family of all 𝑅-osgs (resp., 𝑆-osgs, 𝑃-osgs, Ξ³-osgs, Ξ±-osgs, and Ξ²-osgs) of (𝐺, β„±πΊπ‘š) is denoted by ROπΊπ‘š(𝐺) (resp., SOπΊπ‘š(𝐺), POπΊπ‘š(𝐺), 𝛾OπΊπ‘š(𝐺), 𝛼OπΊπ‘š(𝐺), and 𝛽OπΊπ‘š(𝐺)). All of SOπΊπ‘š(𝐺), POπΊπ‘š(𝐺), 𝛾OπΊπ‘š(𝐺), 𝛼OπΊπ‘š(𝐺), and 𝛽OπΊπ‘š(𝐺) are larger than π’―πΊπ‘š and closed under forming arbitrary union.

The family of all 𝑅-csgs (resp., 𝑆-csgs, 𝑃-csgs, Ξ³-csgs, Ξ±-csgs, and Ξ²-csgs) of (𝐺, β„±πΊπ‘š) is denoted by RCπΊπ‘š(𝐺) (resp., SCπΊπ‘š(𝐺), PCπΊπ‘š(𝐺), 𝛾CπΊπ‘š(𝐺), 𝛼CπΊπ‘š(𝐺), and 𝛽CπΊπ‘š(𝐺)).

The near closure (resp., near interior and near boundary) of a subgraph 𝐻 of 𝐺 in a πΊπ‘š-closure space (𝐺,β„±πΊπ‘š) is denoted by Clπ‘—πΊπ‘š(𝑉(𝐻)) (resp. Intπ‘—πΊπ‘š(𝑉(𝐻)) and Bdπ‘—πΊπ‘š(𝑉(𝐻))) and defined by Clπ‘—πΊπ‘š(𝑉(H))=∩{𝑉(𝐹);𝑉(𝐹)is𝑗-csgand𝑉(𝐻)βŠ†π‘‰(𝐹)},resp.,Intπ‘—πΊπ‘š(𝑉(𝐻))=𝑉(𝐺)βˆ’Clπ‘—πΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻))andBdπ‘—πΊπ‘š(𝑉(𝐻))=Clπ‘—πΊπ‘š(𝑉(𝐻))βˆ’Intπ‘—πΊπ‘šξ€Έ,(𝑉(𝐻))whereπ‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(2.7)

Proposition 2.12 (see [10]). Let (𝐺,β„±πΊπ‘š) be πΊπ‘š-closure space, the implication π’―πΊπ‘š and the families of near-open and near-closed graphs are given by following statements:(a)ROπΊπ‘š(𝐺)βŠ†π’―πΊπ‘šβŠ†π›ΌOπΊπ‘š(𝐺)βŠ†SOπΊπ‘š(𝐺)βŠ†π›ΎOπΊπ‘š(𝐺)βŠ†π›½OπΊπ‘š(𝐺),(b)ROπΊπ‘š(G)βŠ†π’―πΊπ‘šβŠ†π›ΌOπΊπ‘š(G)βŠ†POπΊπ‘š(G)βŠ†π›ΎOπΊπ‘š(G)βŠ†π›½OπΊπ‘š(G),(c)RCπΊπ‘š(𝐺)βŠ†β„±πΊπ‘šβŠ†π›ΌCπΊπ‘š(𝐺)βŠ†SCπΊπ‘š(𝐺)βŠ†π›ΎCπΊπ‘š(𝐺)βŠ†π›½CπΊπ‘š(𝐺), (d)RCπΊπ‘š(𝐺)βŠ†β„±πΊπ‘šβŠ†π›ΌCπΊπ‘š(𝐺)βŠ†PCπΊπ‘š(𝐺)βŠ†π›ΎCπΊπ‘š(𝐺)βŠ†π›½CπΊπ‘š(𝐺).

3. Generalization of Pawlak Approximation Spaces

In this section we will generalize Pawlak’s concepts in the case of general relations. Hence, the approximation space πΊπ‘š=(𝐺,ClπΊπ‘š) with general relation ClπΊπ‘š on 𝐺 (i.e., closure operator ClπΊπ‘š on 𝐺) defines a uniquely πΊπ‘š-closure space (𝐺,β„±πΊπ‘š), where β„±πΊπ‘š is the πΊπ‘š-closure space associated with πΊπ‘š. We will give this hypothesis in the following definition.

Definition 3.1. Let πΊπ‘š=(𝐺,ClπΊπ‘š) be an approximation space, where 𝐺 is a finite and nonempty universe graph, ClπΊπ‘š is a general relation on 𝐺, and β„±πΊπ‘š is the πΊπ‘š-closure space associated with πΊπ‘š. Then the triple π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) is called a πΊπ‘š-closure approximation space.
The following definition introduces the lower and the upper approximations in a πΊπ‘š-closure approximation space π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š).

Definition 3.2. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space and π»βŠ†πΊ. The lower approximation (resp., the upper approximation) of 𝐻 is denoted by 𝐿(𝑉(𝐻))(resp.,π‘ˆ(𝑉(𝐻))) and is defined by 𝐿(𝑉(𝐻))=IntπΊπ‘šξ€·(𝑉(𝐻))resp.,π‘ˆ(𝑉(𝐻))=ClπΊπ‘šξ€Έ.(𝑉(𝐻))(3.1)
The following definition introduces new concepts of definability for a subgraph π»βŠ†πΊ in a πΊπ‘š-closure approximation space π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š).

Definition 3.3. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then 𝐻 is called(a)totally π’’π‘š-definable (π’’π‘š-exact) graph if 𝐿(𝑉(𝐻))=𝑉(𝐻)=π‘ˆ(𝑉(𝐻)),(b)internally π’’π‘š-definable graph if 𝐿(𝑉(𝐻))=𝑉(𝐻),π‘ˆ(𝑉(𝐻))≠𝑉(𝐻),(c)externally π’’π‘š-definable graph if 𝐿(𝑉(𝐻))≠𝑉(𝐻),π‘ˆ(𝑉(𝐻))=𝑉(𝐻),(d)π’’π‘š-indefinable (π’’π‘š-rough) graph if 𝐿(𝑉(𝐻))≠𝑉(𝐻),π‘ˆ(𝑉(𝐻))≠𝑉(𝐻).

Proposition 3.4. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If 𝐻 and 𝐾 are subgraphs of 𝐺, then(1)𝐿(𝑉(𝐻))βŠ†π‘‰(𝐻)βŠ†π‘ˆ(𝑉(𝐻)),(2)𝐿(πœ™)=π‘ˆ(πœ™)=πœ™π‘Žπ‘›π‘‘πΏ(𝑉(𝐺))=π‘ˆ(𝑉(𝐺))=𝑉(𝐺), (3)π‘ˆ(𝑉(𝐻)βˆͺ𝑉(𝐾))=π‘ˆ(𝑉(𝐻))βˆͺπ‘ˆ(𝑉(𝐾)), (4)𝐿(𝑉(𝐻)βˆ©π‘‰(𝐾))=𝐿(𝑉(𝐻))∩𝐿(𝑉(𝐾)), (5)ifπ»βŠ†πΎ,then𝐿(𝑉(𝐻))βŠ†πΏ(𝑉(𝐾)), (6)ifπ»βŠ†πΎ,thenπ‘ˆ(𝑉(𝐻))βŠ†π‘ˆ(𝑉(𝐾)), (7)𝐿(𝑉(𝐻)βˆͺ𝑉(𝐾))βŠ‡πΏ(𝑉(𝐻))βˆͺ𝐿(𝑉(𝐾)), (8)π‘ˆ(𝑉(𝐻)βˆ©π‘‰(𝐾))βŠ†π‘ˆ(𝑉(𝐻))βˆ©π‘ˆ(𝑉(𝐾)), (9)𝐿(𝑉(𝐺)βˆ’π‘‰(𝐻))=𝑉(𝐺)βˆ’π‘ˆ(𝑉(𝐻)),(10)π‘ˆ(𝑉(𝐺)βˆ’π‘‰(𝐻))=𝑉(𝐺)βˆ’πΏ(𝑉(𝐻)).

Proof. By using properties of πΊπ‘š-interior and πΊπ‘š-closure, the proof is obvious.

The following example illustrates that properties 11 and 12 which are introduced in Section 2.1 cannot be applied for this new generalization.

Example 3.5. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space such that 𝐺=(𝑉(𝐺),𝐸(𝐺)):𝑉(𝐺)={𝑣1,𝑣2,𝑣3,𝑣4}, 𝐸(𝐺)={(𝑣2,𝑣1),(𝑣2,𝑣4),(𝑣3,𝑣1),(𝑣4,𝑣1),(𝑣4,𝑣1)},240315.fig.003ℱ𝐺={𝑉(𝐺),πœ™,{𝑣1},{𝑣1,𝑣3},{𝑣1,𝑣2,𝑣4}}, 𝒯𝐺={𝑉(𝐺),πœ™,{𝑣3},{𝑣2,𝑣4},{𝑣2,𝑣3,𝑣4}}. Let 𝐻=(𝑉(𝐻),𝐸(𝐻)): 𝑉(𝐻)={𝑣1,𝑣2,𝑣3},𝐸(𝐻)={(𝑣2,𝑣1),(𝑣3,𝑣1)}, and 𝐾=(𝑉(𝐾),𝐸(𝐾)): 𝑉(𝐾)={𝑣1,𝑣2,𝑣4},𝐸(𝐾)={(𝑣2,𝑣1),(𝑣2,𝑣4),(𝑣4,𝑣1),(𝑣4,𝑣2)}. Then 𝐿𝑣(𝐿(𝑉(𝐻)))=𝐿(𝑉(𝐻))=3𝑣,π‘ˆ(𝐿(𝑉(𝐻)))=1,𝑣3ξ€Ύ.(3.2) Thus, 𝐿(𝐿(𝑉(𝐻)))=𝐿(𝑉(𝐻))β‰ π‘ˆ(𝐿(𝑉(𝐻))).(3.3) Also, π‘ˆξ€½π‘£(π‘ˆ(𝑉(𝐻)))=π‘ˆ(𝑉(𝐻))=1,𝑣2,𝑣4𝑣,𝐿(π‘ˆ(𝑉(𝐻)))=2,𝑣4ξ€Ύ.(3.4) Thus, π‘ˆ(π‘ˆ(𝑉(𝐻)))=π‘ˆ(𝑉(𝐻))≠𝐿(π‘ˆ(𝑉(𝐻))).(3.5)

Lemma 3.6. Let (𝐺,β„±πΊπ‘š) be a πΊπ‘š-closure space. Then IntπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻))=𝑉(𝐺)βˆ’ClπΊπ‘š(𝑉(𝐻))βˆ€subgraphπ»βŠ†πΊ.(3.6)

Proof. It follows from definition of πΊπ‘š-closure space.

Lemma 3.7. Let 𝐻 be a subgraph of 𝐺 in the πΊπ‘š-closure space (𝐺,β„±πΊπ‘š). Then π‘£βˆˆClπΊπ‘š(𝑉(𝐻)) if and only if for each subgraph πΎβŠ†πΊ and π‘£βˆˆIntπΊπ‘š(𝑉(𝐾)), then IntπΊπ‘š(𝑉(𝐾))βˆ©π‘‰(𝐻)β‰ πœ™.

Proof. (β‡’) Let π‘£βˆˆClπΊπ‘š(𝑉(𝐻)) and π‘£βˆˆIntπΊπ‘š(𝑉(𝐾)) for some πΎβŠ†πΊ. Assume IntπΊπ‘š(𝑉(𝐾))βˆ©π‘‰(𝐻)=πœ™. This implies that 𝑉(𝐻)βŠ†π‘‰(𝐺)βˆ’IntπΊπ‘š(𝑉(𝐾)) which is closed graph. Hence, π‘£βˆˆπ‘‰(𝐺)βˆ’IntπΊπ‘š(𝑉(𝐾)), since π‘£βˆˆClπΊπ‘š(𝑉(𝐻)) and this leads to a contradiction. Therefore, IntπΊπ‘š(𝑉(𝐾))βˆ©π‘‰(𝐻)β‰ πœ™.
(⇐) Suppose that for each πΎβŠ†πΊ and π‘£βˆˆIntπΊπ‘š(𝑉(𝐾)), IntπΊπ‘š(𝑉(𝐾))βˆ©π‘‰(𝐻)β‰ πœ™. Let π‘£βˆ‰ClπΊπ‘š(𝑉(𝐻)) which is closed. Then there exists a closed graph πΉβŠ†πΊπ‘ uch that πΉβŠ‡π» and π‘£βˆ‰π‘‰(𝐹). Hence, 𝑉(𝐺)βˆ’π‘‰(𝐹) is open subgraph containing 𝑣. Thus, π‘£βˆˆIntπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐹))=𝑉(𝐺)βˆ’π‘‰(𝐹) and IntπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐹))βˆ©π‘‰(𝐻)=πœ™, that is, there exists a subgraph 𝐾=πΊβˆ’πΉ of 𝐺 such that IntπΊπ‘š(𝑉(𝐾))βˆ©π‘‰(𝐻)=πœ™, which leads to a contradiction. Therefore, π‘£βˆˆClπΊπ‘š(𝑉(𝐻).

Lemma 3.8. Let 𝐻 and 𝐾 be two subgraphs of 𝐺 in the πΊπ‘š-closure space (𝐺,β„±πΊπ‘š). If 𝐻 is open subgraph, then 𝑉(𝐻)∩ClπΊπ‘š(𝑉(𝐾))βŠ†ClπΊπ‘š(𝑉(𝐻)βˆ©π‘‰(𝐾)).

Proof. Let π‘£βˆˆπ‘‰(𝐻)∩ClπΊπ‘š(𝑉(𝐾)). If 𝑂 is open subgraph such that π‘£βˆˆπ‘‰(𝑂), then 𝑉(𝑂)βˆ©π‘‰(𝐻) is an open subgraph and π‘£βˆˆπ‘‰(𝑂)βˆ©π‘‰(𝐻). Therefore, 𝑉(𝑂)∩(𝑉(𝐻)βˆ©π‘‰(𝐾))β‰ πœ™andπ‘£βˆˆClπΊπ‘š(𝑉(𝐻)βˆ©π‘‰(𝐾)). Hence, the result.

Proposition 3.9. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If 𝐻 and 𝐾 are subgraphs of 𝐺, then(a)𝐿(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†πΏ(𝑉(𝐻))βˆ’πΏ(𝑉(𝐾)), (b)π‘ˆ(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ‡π‘ˆ(𝑉(𝐻)βˆ’π‘ˆ(𝑉(𝐾)).

Proof. (a) We need to show that IntπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†IntπΊπ‘š(𝑉(𝐻))βˆ’IntπΊπ‘š(𝑉(𝐾)). Now, 𝑉(𝐻)βˆ’π‘‰(𝐾)=𝑉(𝐻)∩(𝑉(𝐺)βˆ’π‘‰(𝐾)).(3.7) Then, IntπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))=IntπΊπ‘š=(𝑉(𝐻)∩(𝑉(𝐺)βˆ’π‘‰(𝐾)))IntπΊπ‘š(𝑉(𝐻))∩IntπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐾)).(3.8) Thus, by Lemma 3.6, we have IntπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))=IntπΊπ‘šξ€·π‘‰(𝑉(𝐻))∩(𝐺)βˆ’ClπΊπ‘šξ€Έ=(𝑉(𝐾))IntπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘šβŠ†(𝑉(𝐾))IntπΊπ‘š(𝑉(𝐻))βˆ’IntπΊπ‘š(𝑉(𝐾)).(3.9) Therefore, 𝐿(𝑉(𝐻)βˆ’π‘‰(𝐾))=IntπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†IntπΊπ‘š(𝑉(𝐻))βˆ’IntπΊπ‘š(𝑉(𝐾))=𝐿(𝑉(𝐻))βˆ’πΏ(𝑉(𝐾)).(3.10)(b) We need to show that ClπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ‡ClπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘š(𝑉(𝐾)).(3.11) Now, ClπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘š(𝑉(𝐾))=ClπΊπ‘šξ€·π‘‰(𝑉(𝐻))∩(𝐺)βˆ’ClπΊπ‘šξ€Έ.(𝑉(𝐾))(3.12) Thus, by Lemma 3.6, we have ClπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘š(𝑉(𝐾))=ClπΊπ‘š(𝑉(𝐻))∩IntπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐾)).(3.13) Hence, by Lemma 3.8, we have ClπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘š(𝑉(𝐾))=ClπΊπ‘š(𝑉(𝐻))∩IntπΊπ‘šβŠ†(𝑉(𝐺)βˆ’π‘‰(𝐾))ClπΊπ‘šξ€Ίπ‘‰(𝐻)∩IntπΊπ‘šξ€»=(𝑉(𝐺)βˆ’π‘‰(𝐾))ClπΊπ‘šξ€Ίπ‘‰(𝐻)βˆ©π‘‰(𝐺)βˆ’ClπΊπ‘šξ€»=(𝑉(𝐾))ClπΊπ‘šξ€Ίπ‘‰(𝐻)βˆ’ClπΊπ‘šξ€»,(𝑉(𝐾))(3.14) Thus, ClπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘šβŠ†(𝑉(𝐾))ClπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾)).(3.15) Therefore, π‘ˆ(𝑉(𝐻)βˆ’π‘‰(𝐾))=ClπΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ‡ClπΊπ‘š(𝑉(𝐻))βˆ’ClπΊπ‘š(𝑉(𝐾))=π‘ˆ(𝑉(𝐻))βˆ’π‘ˆ(𝑉(𝐾)).(3.16)

4. Near Lower and Near Upper in πΊπ‘š-Closure Approximation Spaces

In this section, we study approximation spaces from πΊπ‘š-closure view. We obtain some rules to find lower and upper approximations in several ways in approximation spaces with general relations. We will recall and introduce some definitions and propositions about some classes of near-open graphs which are essential for our present study.

Definition 4.1. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space and π»βŠ†πΊ. The near-lower approximation (𝑗-lower approximation) (resp., near-upper approximation (𝑗-upper approximation)) of 𝐻 is denoted by 𝐿𝑗(𝑉(𝐻))(resp.,π‘ˆπ‘—(𝑉(𝐻))) and is defined by 𝐿𝑗(𝑉(𝐻))=Intπ‘—πΊπ‘šξ€·(𝑉(𝐻))resp.,π‘ˆπ‘—(𝑉(𝐻))=Clπ‘—πΊπ‘šξ€Έ,(𝑉(𝐻))whereπ‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(4.1)

Proposition 4.2. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then 𝐿(𝑉(𝐻))βŠ†πΏπ‘—(𝑉(𝐻))βŠ†π‘‰(𝐻)βŠ†π‘ˆπ‘—(𝑉(𝐻))βŠ†π‘ˆ(𝑉(𝐻)), for all π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽}.

Proof. The proofs of the five cases are similar, so we will only prove the case when 𝑗=𝑆. Now, π‘ˆ(𝑉(𝐻))=ClπΊπ‘šξ€½π‘‰(𝑉(𝐻))=∩(𝐹);𝑉(𝐹)βˆˆβ„±πΊπ‘šand𝑉(𝐻)βŠ†π‘‰(𝐹)βŠ‡βˆ©π‘‰(𝐹);𝑉(𝐹)∈SCπΊπ‘š(𝐺)and𝑉(𝐻)βŠ†π‘‰(𝐹)sinceβ„±πΊπ‘šβŠ†SCπΊπ‘š=(𝐺)Clπ‘†πΊπ‘š(𝑉(𝐻))=π‘ˆπ‘†(𝑉(𝐻))βŠ‡π‘‰(𝐻),(4.2)𝐿(𝑉(𝐻))=IntπΊπ‘š(𝑉(𝐻))=𝑉(𝐺)βˆ’ClπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻))βŠ†π‘‰(𝐺)βˆ’Clπ‘†πΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻))sinceπ’―πΊπ‘šβŠ†SOπΊπ‘š=(𝐺)Intπ‘†πΊπ‘š(𝑉(𝐻))=𝐿𝑆(𝑉(𝐻))βŠ†π‘‰(𝐻).(4.3) From (4.2) and (4.3) we get 𝐿(𝑉(𝐻))βŠ†πΏπ‘†(𝑉(𝐻))βŠ†π‘‰(𝐻)βŠ†π‘ˆπ‘†(𝑉(𝐻))βŠ†π‘ˆ(𝑉(𝐻)).

In general the above proposition is not true in the case of 𝑗=𝑅 as the following example illustrates.

Example 4.3. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space such that 𝐺=(𝑉(𝐺),𝐸(𝐺)): 𝑉(𝐺)={𝑣1,𝑣2,𝑣3},𝐸(𝐺)={(𝑣2,𝑣1),(𝑣2,𝑣3)},240315.fig.004ℱ𝐺={𝑉(𝐺),πœ™,{𝑣1},{𝑣3},{𝑣1,𝑣3}}, 𝒯𝐺={𝑉(𝐺),πœ™,{𝑣2},{𝑣1,𝑣2},{𝑣2,𝑣3}}. Hence, ROπΊπ‘š(𝐺)={𝑉(𝐺),πœ™}andRCπΊπ‘š(𝐺)={𝑉(𝐺),πœ™}. If 𝐻=(𝑉(𝐻),𝐸(𝐻)): 𝑉(𝐻)={𝑣1,𝑣3},𝐸(𝐻)=πœ™, then 𝐿(𝑉(𝐻))=IntπΊπ‘š(𝑉(𝐻))=πœ™,π‘ˆ(𝑉(𝐻))=ClπΊπ‘šξ€½π‘£(𝑉(𝐻))=1,𝑣3ξ€Ύ,𝐿𝑅(𝑉(𝐻))=Intπ‘…πΊπ‘š(𝑉(𝐻))=πœ™,π‘ˆπ‘…(𝑉(𝐻))=Clπ‘…πΊπ‘š(𝑉(𝐻))=𝑉(𝐺).(4.4) Therefore, 𝐿𝑅(𝑉(𝐻))=𝐿(𝑉(𝐻)),π‘ˆ(𝑉(𝐻))βŠ†π‘ˆπ‘…(𝑉(𝐻)).(4.5)

Proposition 4.4. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a Gm- closure approximation space. If π»βŠ†πΊ, then the implication between lower approximation and 𝑗-lower approximation of 𝐻 are given by the following statement for all π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽}:(a)𝐿(𝑉(𝐻))βŠ†πΏπ›Ό(𝑉(𝐻))βŠ†πΏπ‘†(𝑉(𝐻))βŠ†πΏπ›Ύ(𝑉(𝐻))βŠ†πΏπ›½(𝑉(𝐻)), (b)𝐿(𝑉(𝐻))βŠ†πΏπ›Ό(𝑉(𝐻))βŠ†πΏπ‘ƒ(𝑉(𝐻))βŠ†πΏπ›Ύ(𝑉(𝐻))βŠ†πΏπ›½(𝑉(𝐻)).

Proof. By using Proposition 4.2, we get 𝐿(𝑉(𝐻))βŠ†πΏπ›Ό(𝑉(𝐻)). We will prove 𝐿𝛼(𝑉(𝐻))βŠ†πΏπ‘†(𝑉(𝐻)). Now, 𝐿𝛼(𝑉(𝐻))=Intπ›ΌπΊπ‘š(𝑉(𝐻))=𝑉(𝐺)βˆ’Clπ›ΌπΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻))βŠ†π‘‰(𝐺)βˆ’Clπ‘†πΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻)),(4.6) since 𝛼OπΊπ‘š(𝐺)βŠ†SOπΊπ‘š(𝐺). Thus, 𝐿𝛼(𝑉(𝐻))=Intπ›ΌπΊπ‘š(𝑉(𝐻))βŠ†Intπ‘†πΊπ‘š(𝑉(𝐻))=𝐿𝑆(𝑉(𝐻)).(4.7) Similarly we can prove the other cases.

Proposition 4.5. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then the implication between upper approximation and 𝑗-upper approximation of 𝐻 are given by the following statement for all π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽},(a)π‘ˆπ›½(𝑉(𝐻))βŠ†π‘ˆπ›Ύ(𝑉(𝐻))βŠ†π‘ˆπ‘†(𝑉(𝐻))βŠ†π‘ˆπ›Ό(𝑉(𝐻))βŠ†π‘ˆ(𝑉(𝐻)), (b)π‘ˆπ›½(𝑉(𝐻))βŠ†π‘ˆπ›Ύ(𝑉(𝐻))βŠ†π‘ˆπ‘ƒ(𝑉(𝐻))βŠ†π‘ˆπ›Ό(𝑉(𝐻))βŠ†π‘ˆ(𝑉(𝐻)).

Proof. By using Proposition 4.2, we get π‘ˆπ›Ό(𝑉(𝐻))βŠ†π‘ˆ(𝑉(𝐻)). We will prove π‘ˆπ‘ƒ(𝑉(𝐻))βŠ†π‘ˆπ›Ό(𝑉(𝐻)). Now, π‘ˆπ‘ƒ(𝑉(𝐻))=Clπ‘ƒπΊπ‘šξ€½(𝑉(𝐻))=βˆ©π‘‰(𝐹);𝑉(𝐹)∈PCπΊπ‘š(𝐺)and𝑉(𝐻)βŠ†π‘‰(𝐹)βŠ†βˆ©π‘‰(𝐹);𝑉(𝐹)βˆˆπ›ΌCπΊπ‘š(𝐺)and𝑉(𝐻)βŠ†π‘‰(𝐹)(4.8) since 𝛼CπΊπ‘š(𝐺)βŠ†PCπΊπ‘š(𝐺). Thus, π‘ˆπ‘ƒ(𝑉(𝐻))=Clπ‘ƒπΊπ‘š(𝑉(𝐻))βŠ†Clπ›ΌπΊπ‘š(𝑉(𝐻))=π‘ˆπ›Ό(𝑉(𝐻)).(4.9) Similarly we can prove the other cases.

Proposition 4.6. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If 𝐻 and 𝐾 are two subgraphs of 𝐺, then, for all π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽},(1)𝐿𝑗(πœ™)=π‘ˆπ‘—(πœ™)=πœ™and𝐿𝑗(𝑉(𝐺))=π‘ˆπ‘—(𝑉(𝐺))=𝑉(𝐺), (2)if𝑉(𝐻)βŠ†π‘‰(𝐾),then𝐿𝑗(𝑉(𝐻))βŠ†πΏπ‘—(𝑉(𝐾)), (3)if𝑉(𝐻)βŠ†π‘‰(𝐾),thenπ‘ˆπ‘—(𝑉(𝐻))βŠ†π‘ˆπ‘—(𝑉(𝐾)), (4)𝐿𝑗(𝑉(𝐻)βˆͺ𝑉(𝐾))βŠ‡πΏπ‘—(𝑉(𝐻))βˆͺ𝐿𝑗(𝑉(𝐾)), (5)π‘ˆπ‘—(𝑉(𝐻)βˆͺ𝑉(𝐾))βŠ‡π‘ˆπ‘—(𝑉(𝐻))βˆͺπ‘ˆπ‘—(𝑉(𝐾)), (6)𝐿𝑗(𝑉(𝐻)βˆ©π‘‰(𝐾))βŠ†πΏπ‘—(𝑉(𝐻))βˆ©πΏπ‘—(𝑉(𝐾)), (7)π‘ˆπ‘—(𝑉(𝐻)βˆ©π‘‰(𝐾))βŠ†π‘ˆπ‘—(𝑉(𝐻))βˆ©π‘ˆπ‘—(𝑉(𝐾)), (8)𝐿𝑗(𝑉(𝐻)𝑐)=[π‘ˆπ‘—(𝑉(𝐻))]𝑐, (9)π‘ˆπ‘—(𝑉(𝐻)𝑐)=[𝐿𝑗(𝑉(𝐻))]𝑐.

Proof. By using properties of 𝑗-interior and 𝑗-closure for all π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}, the proof is obvious.

In general, properties 3 and 4 which are introduced in Section 2.1 cannot be applied for 𝑗-lower and 𝑗-upper approximations, where π‘—βˆˆ{𝑆,𝑃,𝛾,𝛽}. The following example illustrates this fact in the case of 𝑗=𝛽.

Example 4.7. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space which is given in Example 2.3:ℱ𝐺={𝑉(𝐺),πœ™,{𝑣3},{𝑣3,𝑣4},{𝑣1,𝑣2,𝑣3}}, 𝒯𝐺={𝑉(𝐺),πœ™,{𝑣4},{𝑣1,𝑣2},{𝑣1,𝑣2,𝑣4}}. 𝛽O𝐺1𝑉𝑣(𝐺)=(𝐺),πœ™,1ξ€Ύ,𝑣2ξ€Ύ,𝑣4ξ€Ύ,𝑣1,𝑣2ξ€Ύ,𝑣1,𝑣3𝑣1,𝑣4ξ€Ύ,𝑣2,𝑣3ξ€Ύ,𝑣2,𝑣4ξ€Ύ,𝑣3,𝑣4ξ€Ύ,𝑣1,𝑣2,𝑣3ξ€Ύ,𝑣1,𝑣2,𝑣4ξ€Ύ,𝑣1,𝑣3,𝑣4ξ€Ύ,𝑣2,𝑣3,𝑣4,𝛽C𝐺1𝑉𝑣(𝐺)=(𝐺),πœ™,1ξ€Ύ,𝑣2ξ€Ύ,𝑣3ξ€Ύ,𝑣4ξ€Ύ,𝑣1,𝑣2𝑣1,𝑣3ξ€Ύ,𝑣1,𝑣4ξ€Ύ,𝑣2,𝑣3ξ€Ύ,𝑣2,𝑣4ξ€Ύ,𝑣3,𝑣4ξ€Ύ,𝑣1,𝑣2,𝑣3ξ€Ύ,𝑣1,𝑣3,𝑣4ξ€Ύ,𝑣2,𝑣3,𝑣4.ξ€Ύξ€Ύ(4.10) If 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻));𝑉(𝐻)=1,𝑣3𝑣,𝐸(𝐻)=ξ€½ξ€·1,𝑣3,𝑣𝐾=(𝑉(𝐾),𝐸(𝐾));𝑉(𝐾)=2,𝑣3𝑣,𝐸(𝐾)=ξ€½ξ€·2,𝑣3,ξ€Έξ€Ύ(4.11) then 𝐿𝛽(𝑉(𝐻))βˆ©πΏπ›½ξ€½π‘£(𝑉(𝐾))=1,𝑣3ξ€Ύβˆ©ξ€½π‘£2,𝑣3ξ€Ύ=𝑣3ξ€Ύ,(4.12) but 𝐿𝛽(𝑉(𝐻)βˆ©π‘‰(𝐾))=πœ™.(4.13) Thus, 𝐿𝛽(𝑉(𝐻)βˆ©π‘‰(𝐾))≠𝐿𝛽(𝑉(𝐻))βˆ©πΏπ›½(𝑉(𝐾)).(4.14) Also, if 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻));𝑉(𝐻)=1,𝑣2𝑣,𝐸(𝐻)=ξ€½ξ€·1,𝑣2ξ€Έ,𝑣2,𝑣1,𝑣𝐾=(𝑉(𝐾),𝐸(𝐾));𝑉(𝐾)=1,𝑣4ξ€Ύ,𝐸(𝐾)=πœ™,(4.15) then π‘ˆπ›½(𝑉(𝐻))βˆͺπ‘ˆπ›½ξ€½π‘£(𝑉(𝐾))=1,𝑣2ξ€Ύβˆͺ𝑣1,𝑣4ξ€Ύ=𝑣1,𝑣2,𝑣4ξ€Ύ,(4.16) but π‘ˆπ›½(𝑉(𝐻)βˆͺ𝑉(𝐾))=𝑉(𝐺).(4.17) Thus, π‘ˆπ›½(𝑉(𝐻)βˆͺ𝑉(𝐾))β‰ π‘ˆπ›½(𝑉(𝐻))βˆͺ𝐿𝛽(𝑉(𝐾)).(4.18)

In general, properties 11 and 12 which are introduced in Section 2.1 cannot be applied for 𝑗-lower and 𝑗-upper approximations, where π‘—βˆˆ{𝑆,𝑃,𝛾,𝛽}. The following example illustrates this fact in the case of 𝑗=𝛽.

Example 4.8. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space which is given in Example 2.3. If 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻));𝑉(𝐻)=1,𝑣2,𝑣4𝑣,𝐸(𝐻)=ξ€½ξ€·1,𝑣2ξ€Έ,𝑣2,𝑣1,𝑣𝐾=(𝑉(𝐾),𝐸(𝐾));𝑉(𝐾)=3ξ€Ύ,𝐸(𝐾)=πœ™,(4.19) then 𝐿𝛽𝐿𝛽(𝑉(𝐻))=𝐿𝛽𝑣(𝑉(𝐻))=1,𝑣2,𝑣4ξ€Ύ,π‘ˆπ›½ξ€·πΏπ›½ξ€Έ(𝑉(𝐻))=𝑉(𝐺).(4.20) Thus, 𝐿𝛽𝐿𝛽(𝑉(𝐻))=𝐿𝛽(𝑉(𝐻))β‰ π‘ˆπ›½ξ€·πΏπ›½ξ€Έ.(𝑉(𝐻))(4.21) Also, π‘ˆπ›½ξ€·π‘ˆπ›½ξ€Έ(𝑉(𝐾))=π‘ˆπ›½ξ€½π‘£(𝑉(𝐾))=3ξ€Ύ,πΏπ›½ξ€·π‘ˆπ›½ξ€Έ(𝑉(𝐾))=πœ™.(4.22) Hence, π‘ˆπ›½ξ€·π‘ˆπ›½ξ€Έ(𝑉(𝐾))=π‘ˆπ›½(𝑉(𝐾))β‰ πΏπ›½ξ€·π‘ˆπ›½ξ€Έ.(𝑉(𝐾))(4.23)

Lemma 4.9. Let (𝐺,β„±πΊπ‘š) be a πΊπ‘š-closure space. Then Intπ‘—πΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐻))=𝑉(𝐺)βˆ’Clπ‘—πΊπ‘š(𝑉(𝐻)) for all subgraph π»βŠ†πΊ and π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.

Proof. It follows from definition near-open subgraphs in Gm-closure space.

Proposition 4.10. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If 𝐻 and 𝐾 are subgraphs of 𝐺, then 𝐿𝑗(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†πΏπ‘—(𝑉(𝐻))βˆ’πΏπ‘—(𝑉(𝐾)),βˆ€π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(4.24)

Proof. We need to show that Intπ‘—πΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†Intπ‘—πΊπ‘š(𝑉(𝐻))βˆ’Intπ‘—πΊπ‘š(𝑉(𝐾)).(4.25) Now, 𝑉(𝐻)βˆ’π‘‰(𝐾)=𝑉(𝐻)∩(𝑉(𝐺)βˆ’π‘‰(𝐾)).(4.26) Then Intπ‘—πΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))=Intπ‘—πΊπ‘šβŠ†(𝑉(𝐻)∩(𝑉(𝐺)βˆ’π‘‰(𝐾)))Intπ‘—πΊπ‘š(𝑉(𝐻))∩Intπ‘—πΊπ‘š(𝑉(𝐺)βˆ’π‘‰(𝐾)).(4.27) Thus, by Lemma 4.9, we have Intπ‘—πΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†Intπ‘—πΊπ‘šξ€·(𝑉(𝐻))βˆ©π‘‰(𝐺)βˆ’Clπ‘—πΊπ‘šξ€Έ=(𝑉(𝐾))Intπ‘—πΊπ‘š(𝑉(𝐻))βˆ’Clπ‘—πΊπ‘š(𝑉(𝐾))βŠ†Intπ‘—πΊπ‘š(𝑉(𝐻))βˆ’Intπ‘—πΊπ‘š(𝑉(𝐾)).(4.28) Therefore 𝐿𝑗(𝑉(𝐻)βˆ’π‘‰(𝐾))=Intπ‘—πΊπ‘š(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†Intπ‘—πΊπ‘š(𝑉(𝐻))βˆ’Intπ‘—πΊπ‘š(𝑉(𝐾))=𝐿𝑗(𝑉(𝐻))βˆ’πΏπ‘—(𝑉(𝐾)).(4.29)

In general, part (b) in Proposition 3.9 cannot be applied for 𝑗-upper approximations for all π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}. Example 4.11 (resp., Example 4.12) illustrates that part (b) in Proposition 3.9 cannot be applied in the case of 𝑗=𝛽(resp.,𝑗=𝑅).

Example 4.11. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space which is given in Example 2.3. If 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻));𝑉(𝐻)=1,𝑣2,𝑣4𝑣,𝐸(𝐻)=ξ€½ξ€·1,𝑣2ξ€Έ,𝑣2,𝑣1,𝑣𝐾=(𝑉(𝐾),𝐸(𝐾));𝑉(𝐾)=1,𝑣2𝑣,𝐸(𝐾)=ξ€½ξ€·1,𝑣2ξ€Έ,𝑣2,𝑣1,ξ€Έξ€Ύ(4.30) then π‘ˆπ›½(𝑉(𝐻)βˆ’π‘‰(𝐾))=π‘ˆπ›½π‘£ξ€·ξ€½4=𝑣4ξ€Ύ,(4.31) but π‘ˆπ›½(𝑉(𝐻))βˆ’π‘ˆπ›½ξ€½π‘£(𝑉(𝐾))=𝑉(𝐺)βˆ’1,𝑣2ξ€Ύ=𝑣3,𝑣4ξ€Ύ.(4.32) Hence, π‘ˆπ›½(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†π‘ˆπ›½(𝑉(𝐻))βˆ’π‘ˆπ›½(𝑉(𝐾)).(4.33)

Example 4.12. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space which is given in Example 2.3: RO𝐺1𝑉𝑣(𝐺)=(𝐺),πœ™,4ξ€Ύ,𝑣1,𝑣2,ξ€Ύξ€ΎRC𝐺1(𝑣𝐺)=𝑉(𝐺),πœ™,3,𝑣4ξ€Ύ,𝑣1,𝑣2,𝑣3.ξ€Ύξ€Ύ(4.34) If 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻));𝑉(𝐻)=1𝑣,𝐸(𝐻)=πœ™,𝐾=(𝑉(𝐾),𝐸(𝐾));𝑉(𝐾)=3ξ€Ύ,𝐸(𝐾)=πœ™,(4.35) then π‘ˆπ‘…(𝑉(𝐻)βˆ’π‘‰(𝐾))=π‘ˆπ‘…(πœ™)=πœ™,(4.36) but π‘ˆπ‘…(𝑉(𝐻))βˆ’π‘ˆπ‘…ξ€½π‘£(𝑉(𝐾))=1,𝑣2,𝑣3ξ€Ύβˆ’ξ€½π‘£3ξ€Ύ=𝑣1,𝑣2ξ€Ύ.(4.37) Hence, π‘ˆπ‘…(𝑉(𝐻)βˆ’π‘‰(𝐾))βŠ†π‘ˆπ‘…(𝑉(𝐻))βˆ’π‘ˆπ‘…(𝑉(𝐾)).(4.38)

5. Near-Boundary Regions and Near Accuracy in πΊπ‘š-Closure Approximation Spaces

In this section we divide the boundary region into several levels. These levels help to decrease the boundary region. In the following definition we introduce the near boundary region of a subgraph 𝐻 of 𝐺 in a πΊπ‘š-closure approximation space π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š).

Definition 5.1. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space and π»βŠ†πΊ. The near-boundary (𝑗-boundary) region of 𝐻 is denoted by Bdπ‘—π’’π‘š(𝑉(𝐻)) and is defined by Bdπ‘—π’’π‘š(𝑉(𝐻))=π‘ˆπ‘—(𝑉(𝐻))βˆ’πΏπ‘—(𝑉(𝐻)),whereπ‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(5.1)

Definition 5.2. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space and π»βŠ†πΊ. The near-positive (𝑗-positive) region of 𝐻 is denoted by POSπ‘—π’’π‘š(𝑉(𝐻)) and is defined by POSπ‘—π’’π‘š(𝑉(𝐻))=𝐿𝑗(𝑉(𝐻)),whereπ‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(5.2)

Definition 5.3. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space and π»βŠ†πΊ. The near negative (briefly 𝑗-negative) region of 𝐻 is denoted by NEGπ‘—π’’π‘š(𝑉(𝐻)) and is defined by NEGπ‘—π’’π‘š(𝑉(𝐻))=𝑉(𝐺)βˆ’π‘ˆπ‘—(𝑉(𝐻)),whereπ‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(5.3)

Proposition 5.4. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then Bdπ‘—π’’π‘š(𝑉(𝐻))βŠ†Bdπ’’π‘š(𝑉(𝐻))βˆ€π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽}.(5.4)

Proof. By using Proposition 4.2, the proof is obvious.

In general, the above proposition is not true in the case of 𝑗=𝑅 as illustrated in the following example.

Example 5.5. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space which is given in Example 2.3. If 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻))βˆΆπ‘‰(𝐻)=1,𝑣3𝑣,𝐸(𝐻)=ξ€½ξ€·1,𝑣3,ξ€Έξ€Ύ(5.5) then Bdπ’’π‘š(𝑉(𝐻))=π‘ˆ(𝑉(𝐻))βˆ’πΏ(𝑉(𝐻))=ClπΊπ‘š(𝑉(𝐻))βˆ’IntπΊπ‘š=(𝑉(𝐻))ClπΊπ‘šπ‘£ξ€·ξ€½1,𝑣3βˆ’ξ€Ύξ€ΈIntπΊπ‘šπ‘£ξ€·ξ€½1,𝑣3=𝑣1,𝑣3ξ€Ύξ€½π‘£βˆ’πœ™=1,𝑣3ξ€Ύ,Bdπ‘…π’’π‘š(𝑉(𝐻))=π‘ˆπ‘…(𝑉(𝐻))βˆ’πΏπ‘…(𝑉(𝐻))=Clπ‘…π’’π‘š(𝑉(𝐻))βˆ’Intπ‘…π’’π‘š=(𝑉(𝐻))Clπ‘…π’’π‘šπ‘£ξ€·ξ€½1,𝑣3βˆ’ξ€Ύξ€ΈIntπ‘…π’’π‘šπ‘£ξ€·ξ€½1,𝑣3ξ€Ύξ€Έ=𝑉(𝐺)βˆ’πœ™=𝑉(𝐺).(5.6)

Proposition 5.6. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then the implication between boundary and 𝑗-boundary of 𝐻 given by the following statement for all π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽}:(a)Bdπ›½π’’π‘š(𝑉(𝐻))βŠ†Bdπ›Ύπ’’π‘š(𝑉(𝐻))βŠ†Bdπ‘†π’’π‘š(𝑉(𝐻))βŠ†Bdπ›Όπ’’π‘š(𝑉(𝐻))βŠ†Bdπ’’π‘š(𝑉(𝐻)), (b)Bdπ›½π’’π‘š(𝑉(𝐻))βŠ†Bdπ›Ύπ’’π‘š(𝑉(𝐻))βŠ†Bdπ‘ƒπ’’π‘š(𝑉(𝐻))βŠ†Bdπ›Όπ’’π‘š(𝑉(𝐻))βŠ†Bdπ’’π‘š(𝑉(𝐻)).

Proof. By using Propositions 4.4 and 4.5, the proof is obvious.

Definition 5.7. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space and 𝐻 a finite nonempty subgraph of 𝐺. The near accuracy (𝑗-accuracy) of 𝐻 is denoted by πœ‚π‘—π’’π‘š(𝑉(𝐻)) and is defined by πœ‚π‘—π’’π‘š||𝐿(𝑉(𝐻))=𝑗||(𝑉(𝐻))||π‘ˆπ‘—(||,𝑉(𝐻))where||π‘ˆπ‘—||(𝑉(𝐻))β‰ 0βˆ€π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.(5.7)

Proposition 5.8. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If 𝐻 is a finite nonempty subgraph of 𝐺, then πœ‚π’’π‘š(𝑉(𝐻))β‰€πœ‚π‘—π’’π‘š(𝑉(𝐻)) for all π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽}, where πœ‚π’’π‘š(𝑉(𝐻))=|𝐿(𝑉(𝐻))|/|π‘ˆ(𝑉(𝐻))|is the accuracy of 𝐻.

Proof. By using Proposition 4.2, the proof is obvious.

In general, the above proposition is not true in the case of 𝑗=𝑅. This fact is illustrated in the following example.

Example 5.9. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space which is given in Example 4.3. If 𝑣𝐻=(𝑉(𝐻),𝐸(𝐻))βˆΆπ‘‰(𝐻)=1,𝑣2𝑣,𝐸(𝐻)=ξ€½ξ€·2,𝑣1,ξ€Έξ€Ύ(5.8) then πœ‚π’’π‘š2(𝑉(𝐻))=3,πœ‚π‘…π’’π‘š(𝑉(𝐻))=0.(5.9) Thus, πœ‚π‘…π’’π‘š(𝑉(𝐻))<πœ‚π’’π‘š(𝑉(𝐻)).(5.10)

Proposition 5.10. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then the implication between accuracy and 𝑗-accuracy of 𝐻 is given by the following statement for all π‘—βˆˆ{𝑆,𝑃,𝛾,𝛼,𝛽}:(a)πœ‚π’’π‘š(𝑉(𝐻))β‰€πœ‚π›Όπ’’π‘š(𝑉(𝐻))β‰€πœ‚π‘†π’’π‘š(𝑉(𝐻))β‰€πœ‚π›Ύπ’’π‘š(𝑉(𝐻))β‰€πœ‚π›½π’’π‘š(𝑉(𝐻)), (b)πœ‚π’’π‘š(𝑉(𝐻))β‰€πœ‚π›Όπ’’π‘š(𝑉(𝐻))β‰€πœ‚π‘ƒπ’’π‘š(𝑉(𝐻))β‰€πœ‚π›Ύπ’’π‘š(𝑉(𝐻))β‰€πœ‚π›½π’’π‘š(𝑉(𝐻)).

Proof. By using Propositions 4.4 and 4.5, the proof is obvious.

6. Rough and Near-Rough Cluster Vertices in πΊπ‘š-Closure Approximation Spaces

In this section, we introduce the definitions of definability of graphs, rough cluster vertices and near-rough cluster vertices in approximation spaces with general relations. The following definition introduces new concepts of definability for a subgraph π»βŠ†πΊ in a πΊπ‘š-closure approximation space π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š).

Definition 6.1. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. If π»βŠ†πΊ, then 𝐻 is called(a)totally π‘—π’’π‘š-definable (π‘—π’’π‘š-exact) graph if 𝐿𝑗(𝑉(𝐻))=𝑉(𝐻)=π‘ˆπ‘—(𝑉(𝐻)),(b)internally π‘—π’’π‘š-definable graph if 𝐿𝑗(𝑉(𝐻))=𝑉(𝐻),π‘ˆπ‘—(𝑉(𝐻))≠𝑉(𝐻),(c)externally π‘—π’’π‘š-definable graph if 𝐿𝑗(𝑉(𝐻))≠𝑉(𝐻),π‘ˆπ‘—(𝑉(𝐻))=𝑉(𝐻),(d)π‘—π’’π‘š-indefinable (π‘—π’’π‘š-rough) graph if 𝐿𝑗(𝑉(𝐻))≠𝑉(𝐻),π‘ˆπ‘—(𝑉(𝐻))≠𝑉(𝐻), where π‘—βˆˆ{𝑅,𝑆,𝑃,𝛾,𝛼,𝛽}.

Example 6.2. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space such that 𝐺=(𝑉(𝐺),𝐸(𝐺))βˆΆπ‘‰(𝐺)={𝑣1,𝑣2,𝑣3,𝑣4},𝐸(𝐺)={(𝑣2,𝑣1),(𝑣2,𝑣4),(𝑣3,𝑣1),(𝑣4,𝑣1),(𝑣4,𝑣2)}, 240315.fig.005ℱ𝐺={𝑉(𝐺),πœ™,{𝑣1},{𝑣1,𝑣3},{𝑣1,𝑣2,𝑣4}}, 𝒯𝐺={𝑉(𝐺),πœ™,{𝑣3},{𝑣2,𝑣4},{𝑣2,𝑣3,𝑣4}}. Let 𝐻=(𝑉(𝐻),𝐸(𝐻)): 𝑉(𝐻)={𝑣1,𝑣2,𝑣3}, 𝐸(𝐻)={(𝑣2,𝑣1),(𝑣3,𝑣1)}, then, for π‘—βˆˆ{𝑆,𝑃}, we get POSπ‘†π’’π‘š(𝑉(𝐻))=𝐿𝑆(𝑉(𝐻))=Intπ‘†πΊπ‘šξ€½π‘£(𝑉(𝐻))=1,𝑣3ξ€Ύ,π‘ˆπ‘†(𝑉(𝐻))=Clπ‘†πΊπ‘š(𝑉(𝐻))=𝑉(𝐺),Bdπ‘†πΊπ‘š(𝑉(𝐻))=Bdπ‘†π’’π‘šξ€½π‘£(𝑉(𝐻))=2,𝑣4ξ€Ύ,NEGπ‘†π’’π‘š(𝑉(𝐻))=πœ™,POSπ‘ƒπ’’π‘š(𝑉(𝐻))=𝐿𝑃(𝑉(𝐻))=Intπ‘ƒπΊπ‘šξ€½π‘£(𝑉(𝐻))=1,𝑣2,𝑣3ξ€Ύ,π‘ˆπ‘ƒ(𝑉(𝐻))=Clπ‘ƒπΊπ‘š(𝑣𝑉(𝐻))=1,𝑣2,𝑣3ξ€Ύ,Bdπ‘†πΊπ‘š(𝑉(𝐻))=Bdπ‘ƒπ’’π‘š(𝑉(𝐻))=πœ™,NEGπ‘ƒπ’’π‘šξ€½π‘£(𝑉(𝐻))=4ξ€Ύ.(6.1) Thus, 𝐻 is an π‘†π’’π‘š-indefinable (π‘†π’’π‘š-rough) graph and π‘ƒπ’’π‘š-definable (π‘ƒπ’’π‘š-exact) graph.
The following definition introduces the concept of rough cluster vertices of a subgraph 𝐻 of 𝐺 in a πΊπ‘š-closure approximation π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š).

Definition 6.3. Let π’’π‘š=(𝐺,ClπΊπ‘š,β„±πΊπ‘š) be a πΊπ‘š-closure approximation space. The vertex π‘£βˆˆπΊ is said to be a rough cluster vertex of a subgraph 𝐻 of 𝐺 if, for all subgraph 𝐾 of