# Near Approximations inGm-Closure Spaces

Authors: M. E. Abd El-Monsef; M. Shokry; Y. Y. Yousif
Journal: ISRN Applied Mathematics
 (2012)

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## 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 theGm-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.


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## Body

## 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 Gm-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 spaceK=(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 Rx⊆X to denote the equivalence class containing x∈X. In the approximation space, we consider two operators, the upper and lower approximations of subsets: let A⊆X, then the lower approximation (resp., the upper approximation) of A is given by(2.1)L(A)={x∈X:Rx⊆A}(resp.,U(A)={x∈X:Rx∩A≠ϕ}).
Boundary, positive, and negative regions are also defined:(2.2)BdR(A)=U(A)-L(A),POSR(A)=L(A),NEGR(A)=X-U(A).In an approximation spaceK=(X,R), if A and B 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)⊆A⊆U(A),(2)
L(ϕ)=U(ϕ)=ϕandL(X)=U(X)=X,(3)
U(A∪B)=U(A)∪U(B),(4)
L(A∩B)=L(A)∩L(B),(5)
IfA⊆B,thenL(A)⊆L(B),(6)
IfA⊆B,thenU(A)⊆U(B),(7)
L(A∪B)⊇L(A)∪L(B),(8)
U(A∩B)⊆U(A)∩U(B),(9)
L(Ac)=[U(A)]c,(10)
U(Ac)=[L(A)]c,(11)
L(L(A))=U(L(A))=L(A),(12)
U(U(A))=L(U(A))=U(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 A⊆X in approximation space  K=(X,R).Definition 2.1.
LetK=(X,R) be an approximation space. The accuracy measure of a subset A⊆X is defined by η(A) and define by
(2.3)η(A)=|L(A)||U(A)|,where|U(A)|≠0.The accuracy measure is also called the accuracy of approximation.


### 2.2. Fundamental Notions ofGm-Closure Spaces

In this section, we introduce the concepts of closure operators on digraphs; several known topological properties on the obtainedGm-closure spaces are studied.Definition 2.2 (see [10, 11]).
LetG=(V(G),E(G)) be a digraph,  P(V(G)) its power set of all subgraphs of G, and ClG:P(V(G))→P(V(G)) a mapping associating with each subgraph  H=(V(H),E(H)); a subgraph  ClG(V(H))⊆V(G) is called the closure subgraph of H such that
(2.4)ClG(V(H))=V(H)∪{v∈V(G)-V(H);hv⃗∈E(G)∀h∈V(H)}.
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)∈ℱGandV(H)⊆V(F)}. The dual of the graph closure operator ClG is the graph interior operator IntG:P(V(G))→P(V(G)) defined by IntG(V(H))=V(G)-ClG(V(G)-V(H)) for all subgraph H⊆G. 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))=∪{V(O);V(O)∈𝒯GandV(O)⊆V(H)}. Then the domain of ClG is equal to the domain of IntG and also ClG(V(H))=V(G)-IntG(V(G)-V(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)-V(H))=V(G)-V(H), or equivalently IntG(V(H))=V(H).Example 2.3.
LetG=(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)ℱG={V(G),ϕ,{v3},{v3,v4},{v1,v2,v3}},𝒯G={V(G),ϕ,{v4},{v1,v2},{v1,v2,v4}}.
We obtain a new definition to construct topological closure spaces fromG-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.Table 1V(H)ClG(V(H))V(H)ClG(V(H))V(G)V(G){v1,v4}V(G)ϕϕ{v2,v3}{v1,v2,v3}{v1}{v1,v2,v3}{v2,v4}V(G){v2}{v1,v2,v3}{v3,v4}{v3,v4}{v3}{v3}{v1,v2,v3}V(G){v4}{v3,v4}{v1,v2,v4}V(G){v1,v2}{v1,v2,v3}{v1,v3,v4}V(G){v1,v3}{v1,v2,v3}{v2,v3,v4}V(G)Definition 2.4 (see [10, 11]).
LetG=(V(G),E(G)) be a digraph and  ClGm:P(V(G))→P(V(G)) an operator such that:(a)
It is calledGm-closure operator if ClGm(V(H))=ClG(ClG(…ClG(V(H)))), m-times, for every subgraph H⊆G,(b)
it is calledGm-topological closure operator if ClGm+1(V(H))=ClGm(V(H)) for all subgraph H⊆G.The space (G,ℱGm) is called Gm-closure space.Example 2.5.
LetG=(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)ℱG2={V(G),ϕ,{v1,v3,v4}},𝒯G2={V(G),ϕ,{v2}}.Table 2V(H)ClG(V(H))ClG2(V(H))V(H)ClG(V(H))ClG2(V(H))V(G)V(G)V(G){v1,v4}{v1,v3,v4}{v1,v3,v4}ϕϕϕ{v2,v3}V(G)V(G){v1}{v1,v3}{v1,v3,v4}{v2,v4}V(G)V(G){v2}{v1,v2,v3}V(G){v3,v4}{v1,v3,v4}{v1,v3,v4}{v3}{v3,v4}{v1,v3,v4}{v1,v2,v3}V(G)V(G){v4}{v1,v4}{v1,v3,v4}{v1,v2,v4}V(G)V(G){v1,v2}{v1,v2,v3}V(G){v1,v3,v4}{v1,v3,v4}{v1,v3,v4}{v1,v3}{v1,v3,v4}{v1,v3,v4}{v2,v3,v4}V(G)V(G)Proposition 2.6 (see [10]).
Let (G,ℱGm) be a Gm-closure space. If H and K are two subgraphs of G such that H⊆K⊆G, then(2.5)ClGm(V(H))⊆ClGm(V(K)),IntGm(V(H))⊆IntGm(V(K)).Proposition 2.7 (see [10]).
Let (G,ℱGm) be a Gm-closure space. If H and K are two subgraphs of G, then(a)
ClGm(V(H)∪V(K))=ClGm(V(H))∪ClGm(V(K)),(b)
IntGm(V(H)∩V(K))=IntGm(V(H))∩IntGm(V(K)).Proposition 2.8 (see [10]).
Let (G, ℱGm) be a Gm-closure space. If H and K are two subgraphs of G, then(a)
ClGm(V(H)∩V(K))⊆ClGm(V(H))∩ClGm(V(K)), and(b)
IntGm(V(H))∪IntGm(V(K))⊆IntGm(V(H)∪V(K)).Remark 2.9.
The converse of Proposition2.8 need not be true in general, as the following example (Example  2.3 in [10]).Definition 2.10 (see [10]).
Let (G,ℱGm) be a Gm-closure space and H⊆G; the boundary of H is denoted by BdGm(V(H)) and is defined by
(2.6)BdGm(V(H))=ClGm(V(H))-IntGm(V(H)).Proposition 2.11 (see [10]).
Let (G,ℱGm) be a Gm-closure space and H⊆G, then(a)
BdGm(V(H))=ClGm(V(H))∩ClGm(V(G)-V(H)),(b)
BdGm(V(H))=BdGm(V(G)-V(H)),(c)
ClGm(V(H))=V(H)∪BdGm(V(H)),(d)
IntGm(V(H))=V(H)-BdGm(V(H)).By a similar way of definitions of regular open set [17], semiopen set [18], preopen set [19], γ-open set [20] (b-open set [21]), α-open set [22], and β-open set [23] (=semi-pre-open set [24]), we introduce the following definitions which are essential for our present study. In Gm-closure space (G, ℱGm) the subgraph H of G is called(a)
regular open subgraph [10] (briefly R-osg) if V(H)=IntGm(ClGm(V(H))),(b)
semiopen subgraph [10] (briefly S-osg) if V(H)⊆ClGm(IntGm(V(H))),(c)
preopen subgraph [10] (briefly P-osg) if V(H)⊆IntGm(ClGm(V(H))),(d)
γ-open subgraph (briefly γ-osg) if V(H)⊆ClGm(IntGm(V(H)))∪IntGm(ClGm(V(H))),(e)
α-open subgraph [10] (briefly α-osg) if V(H)⊆IntGm(ClGm(IntGm(V(H))),(f)
β-open subgraph [10] (briefly β-osg) if V(H)⊆ClGm(IntGm(ClGmV(H))).The complement of anR-osg (resp., S-osg, P-osg, γ-osg, α-osg, and β-osg) is called R-closed subgraph (briefly R-csg) (resp., S-csg, P-csg, γ-csg, α-csg, and β-csg).The family of allR-osgs (resp., S-osgs, P-osgs, γ-osgs, α-osgs, and β-osgs) of (G, ℱGm) is denoted by ROGm(G) (resp., SOGm(G), POGm(G), γOGm(G), αOGm(G), and βOGm(G)). All of SOGm(G), POGm(G), γOGm(G), αOGm(G), and βOGm(G) are larger than 𝒯Gm and closed under forming arbitrary union.The family of allR-csgs (resp., S-csgs, P-csgs, γ-csgs, α-csgs, and β-csgs) of (G, ℱGm) is denoted by RCGm(G) (resp.,  SCGm(G), PCGm(G), γCGm(G), αCGm(G), and βCGm(G)).The near closure (resp., near interior and near boundary) of a subgraphH of G in a Gm-closure space (G,ℱGm) is denoted by ClGmj(V(H)) (resp. IntGmj(V(H)) and BdGmj(V(H)))  and defined by
(2.7)ClGmj(V(H))=∩{V(F);V(F)isj-csgandV(H)⊆V(F)},(resp.,IntGmj(V(H))=V(G)-ClGmj(V(G)-V(H))andBdGmj(V(H))=ClGmj(V(H))-IntGmj(V(H))),wherej∈{R,S,P,γ,α,β}.Proposition 2.12 (see [10]).
Let (G,ℱGm) be Gm-closure space, the implication 𝒯Gm and the families of near-open and near-closed graphs are given by following statements:(a)
ROGm(G)⊆𝒯Gm⊆αOGm(G)⊆SOGm(G)⊆γOGm(G)⊆βOGm(G),(b)
ROGm(G)⊆𝒯Gm⊆αOGm(G)⊆POGm(G)⊆γOGm(G)⊆βOGm(G),(c)
RCGm(G)⊆ℱGm⊆αCGm(G)⊆SCGm(G)⊆γCGm(G)⊆βCGm(G),(d)
RCGm(G)⊆ℱGm⊆αCGm(G)⊆PCGm(G)⊆γCGm(G)⊆βCGm(G).


## 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 spaceK=(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 Rx⊆X to denote the equivalence class containing x∈X. In the approximation space, we consider two operators, the upper and lower approximations of subsets: let A⊆X, then the lower approximation (resp., the upper approximation) of A is given by(2.1)L(A)={x∈X:Rx⊆A}(resp.,U(A)={x∈X:Rx∩A≠ϕ}).
Boundary, positive, and negative regions are also defined:(2.2)BdR(A)=U(A)-L(A),POSR(A)=L(A),NEGR(A)=X-U(A).In an approximation spaceK=(X,R), if A and B 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)⊆A⊆U(A),(2)
L(ϕ)=U(ϕ)=ϕandL(X)=U(X)=X,(3)
U(A∪B)=U(A)∪U(B),(4)
L(A∩B)=L(A)∩L(B),(5)
IfA⊆B,thenL(A)⊆L(B),(6)
IfA⊆B,thenU(A)⊆U(B),(7)
L(A∪B)⊇L(A)∪L(B),(8)
U(A∩B)⊆U(A)∩U(B),(9)
L(Ac)=[U(A)]c,(10)
U(Ac)=[L(A)]c,(11)
L(L(A))=U(L(A))=L(A),(12)
U(U(A))=L(U(A))=U(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 A⊆X in approximation space  K=(X,R).Definition 2.1.
LetK=(X,R) be an approximation space. The accuracy measure of a subset A⊆X is defined by η(A) and define by
(2.3)η(A)=|L(A)||U(A)|,where|U(A)|≠0.The accuracy measure is also called the accuracy of approximation.


## 2.2. Fundamental Notions ofGm-Closure Spaces

In this section, we introduce the concepts of closure operators on digraphs; several known topological properties on the obtainedGm-closure spaces are studied.Definition 2.2 (see [10, 11]).
LetG=(V(G),E(G)) be a digraph,  P(V(G)) its power set of all subgraphs of G, and ClG:P(V(G))→P(V(G)) a mapping associating with each subgraph  H=(V(H),E(H)); a subgraph  ClG(V(H))⊆V(G) is called the closure subgraph of H such that
(2.4)ClG(V(H))=V(H)∪{v∈V(G)-V(H);hv⃗∈E(G)∀h∈V(H)}.
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)∈ℱGandV(H)⊆V(F)}. The dual of the graph closure operator ClG is the graph interior operator IntG:P(V(G))→P(V(G)) defined by IntG(V(H))=V(G)-ClG(V(G)-V(H)) for all subgraph H⊆G. 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))=∪{V(O);V(O)∈𝒯GandV(O)⊆V(H)}. Then the domain of ClG is equal to the domain of IntG and also ClG(V(H))=V(G)-IntG(V(G)-V(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)-V(H))=V(G)-V(H), or equivalently IntG(V(H))=V(H).Example 2.3.
LetG=(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)ℱG={V(G),ϕ,{v3},{v3,v4},{v1,v2,v3}},𝒯G={V(G),ϕ,{v4},{v1,v2},{v1,v2,v4}}.
We obtain a new definition to construct topological closure spaces fromG-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.Table 1V(H)ClG(V(H))V(H)ClG(V(H))V(G)V(G){v1,v4}V(G)ϕϕ{v2,v3}{v1,v2,v3}{v1}{v1,v2,v3}{v2,v4}V(G){v2}{v1,v2,v3}{v3,v4}{v3,v4}{v3}{v3}{v1,v2,v3}V(G){v4}{v3,v4}{v1,v2,v4}V(G){v1,v2}{v1,v2,v3}{v1,v3,v4}V(G){v1,v3}{v1,v2,v3}{v2,v3,v4}V(G)Definition 2.4 (see [10, 11]).
LetG=(V(G),E(G)) be a digraph and  ClGm:P(V(G))→P(V(G)) an operator such that:(a)
It is calledGm-closure operator if ClGm(V(H))=ClG(ClG(…ClG(V(H)))), m-times, for every subgraph H⊆G,(b)
it is calledGm-topological closure operator if ClGm+1(V(H))=ClGm(V(H)) for all subgraph H⊆G.The space (G,ℱGm) is called Gm-closure space.Example 2.5.
LetG=(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)ℱG2={V(G),ϕ,{v1,v3,v4}},𝒯G2={V(G),ϕ,{v2}}.Table 2V(H)ClG(V(H))ClG2(V(H))V(H)ClG(V(H))ClG2(V(H))V(G)V(G)V(G){v1,v4}{v1,v3,v4}{v1,v3,v4}ϕϕϕ{v2,v3}V(G)V(G){v1}{v1,v3}{v1,v3,v4}{v2,v4}V(G)V(G){v2}{v1,v2,v3}V(G){v3,v4}{v1,v3,v4}{v1,v3,v4}{v3}{v3,v4}{v1,v3,v4}{v1,v2,v3}V(G)V(G){v4}{v1,v4}{v1,v3,v4}{v1,v2,v4}V(G)V(G){v1,v2}{v1,v2,v3}V(G){v1,v3,v4}{v1,v3,v4}{v1,v3,v4}{v1,v3}{v1,v3,v4}{v1,v3,v4}{v2,v3,v4}V(G)V(G)Proposition 2.6 (see [10]).
Let (G,ℱGm) be a Gm-closure space. If H and K are two subgraphs of G such that H⊆K⊆G, then(2.5)ClGm(V(H))⊆ClGm(V(K)),IntGm(V(H))⊆IntGm(V(K)).Proposition 2.7 (see [10]).
Let (G,ℱGm) be a Gm-closure space. If H and K are two subgraphs of G, then(a)
ClGm(V(H)∪V(K))=ClGm(V(H))∪ClGm(V(K)),(b)
IntGm(V(H)∩V(K))=IntGm(V(H))∩IntGm(V(K)).Proposition 2.8 (see [10]).
Let (G, ℱGm) be a Gm-closure space. If H and K are two subgraphs of G, then(a)
ClGm(V(H)∩V(K))⊆ClGm(V(H))∩ClGm(V(K)), and(b)
IntGm(V(H))∪IntGm(V(K))⊆IntGm(V(H)∪V(K)).Remark 2.9.
The converse of Proposition2.8 need not be true in general, as the following example (Example  2.3 in [10]).Definition 2.10 (see [10]).
Let (G,ℱGm) be a Gm-closure space and H⊆G; the boundary of H is denoted by BdGm(V(H)) and is defined by
(2.6)BdGm(V(H))=ClGm(V(H))-IntGm(V(H)).Proposition 2.11 (see [10]).
Let (G,ℱGm) be a Gm-closure space and H⊆G, then(a)
BdGm(V(H))=ClGm(V(H))∩ClGm(V(G)-V(H)),(b)
BdGm(V(H))=BdGm(V(G)-V(H)),(c)
ClGm(V(H))=V(H)∪BdGm(V(H)),(d)
IntGm(V(H))=V(H)-BdGm(V(H)).By a similar way of definitions of regular open set [17], semiopen set [18], preopen set [19], γ-open set [20] (b-open set [21]), α-open set [22], and β-open set [23] (=semi-pre-open set [24]), we introduce the following definitions which are essential for our present study. In Gm-closure space (G, ℱGm) the subgraph H of G is called(a)
regular open subgraph [10] (briefly R-osg) if V(H)=IntGm(ClGm(V(H))),(b)
semiopen subgraph [10] (briefly S-osg) if V(H)⊆ClGm(IntGm(V(H))),(c)
preopen subgraph [10] (briefly P-osg) if V(H)⊆IntGm(ClGm(V(H))),(d)
γ-open subgraph (briefly γ-osg) if V(H)⊆ClGm(IntGm(V(H)))∪IntGm(ClGm(V(H))),(e)
α-open subgraph [10] (briefly α-osg) if V(H)⊆IntGm(ClGm(IntGm(V(H))),(f)
β-open subgraph [10] (briefly β-osg) if V(H)⊆ClGm(IntGm(ClGmV(H))).The complement of anR-osg (resp., S-osg, P-osg, γ-osg, α-osg, and β-osg) is called R-closed subgraph (briefly R-csg) (resp., S-csg, P-csg, γ-csg, α-csg, and β-csg).The family of allR-osgs (resp., S-osgs, P-osgs, γ-osgs, α-osgs, and β-osgs) of (G, ℱGm) is denoted by ROGm(G) (resp., SOGm(G), POGm(G), γOGm(G), αOGm(G), and βOGm(G)). All of SOGm(G), POGm(G), γOGm(G), αOGm(G), and βOGm(G) are larger than 𝒯Gm and closed under forming arbitrary union.The family of allR-csgs (resp., S-csgs, P-csgs, γ-csgs, α-csgs, and β-csgs) of (G, ℱGm) is denoted by RCGm(G) (resp.,  SCGm(G), PCGm(G), γCGm(G), αCGm(G), and βCGm(G)).The near closure (resp., near interior and near boundary) of a subgraphH of G in a Gm-closure space (G,ℱGm) is denoted by ClGmj(V(H)) (resp. IntGmj(V(H)) and BdGmj(V(H)))  and defined by
(2.7)ClGmj(V(H))=∩{V(F);V(F)isj-csgandV(H)⊆V(F)},(resp.,IntGmj(V(H))=V(G)-ClGmj(V(G)-V(H))andBdGmj(V(H))=ClGmj(V(H))-IntGmj(V(H))),wherej∈{R,S,P,γ,α,β}.Proposition 2.12 (see [10]).
Let (G,ℱGm) be Gm-closure space, the implication 𝒯Gm and the families of near-open and near-closed graphs are given by following statements:(a)
ROGm(G)⊆𝒯Gm⊆αOGm(G)⊆SOGm(G)⊆γOGm(G)⊆βOGm(G),(b)
ROGm(G)⊆𝒯Gm⊆αOGm(G)⊆POGm(G)⊆γOGm(G)⊆βOGm(G),(c)
RCGm(G)⊆ℱGm⊆αCGm(G)⊆SCGm(G)⊆γCGm(G)⊆βCGm(G),(d)
RCGm(G)⊆ℱGm⊆αCGm(G)⊆PCGm(G)⊆γCGm(G)⊆βCGm(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 spaceGm=(G,ClGm) with general relation ClGm on G (i.e., closure operator ClGm on G) defines a uniquely Gm-closure space (G,ℱGm), where ℱGm is the Gm-closure space associated with  Gm. We will give this hypothesis in the following definition.Definition 3.1.
LetGm=(G,ClGm) be an approximation space, where G is a finite and nonempty universe graph, ClGm is a general relation on G, and ℱGm is the Gm-closure space associated with  Gm. Then the triple 𝒢m=(G,ClGm,ℱGm) is called a Gm-closure approximation space.
The following definition introduces the lower and the upper approximations in aGm-closure approximation space 𝒢m=(G,ClGm,ℱGm).Definition 3.2.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space and  H⊆G. The lower approximation (resp., the upper approximation) of H is denoted by L(V(H))(resp.,U(V(H))) and is defined by
(3.1)L(V(H))=IntGm(V(H))(resp.,U(V(H))=ClGm(V(H))).
The following definition introduces new concepts of definability for a subgraphH⊆G in a Gm-closure approximation space  𝒢m=(G,ClGm,ℱGm).Definition 3.3.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H⊆G, then H is called(a)
totally𝒢m-definable (𝒢m-exact) graph if L(V(H))=V(H)=U(V(H)),(b)
internally𝒢m-definable graph if L(V(H))=V(H),U(V(H))≠V(H),(c)
externally𝒢m-definable graph if L(V(H))≠V(H),U(V(H))=V(H),(d)
𝒢m-indefinable (𝒢m-rough) graph if L(V(H))≠V(H),U(V(H))≠V(H).Proposition 3.4.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H and K are subgraphs of G, then(1)
L(V(H))⊆V(H)⊆U(V(H)),(2)
L(ϕ)=U(ϕ)=ϕandL(V(G))=U(V(G))=V(G),(3)
U(V(H)∪V(K))=U(V(H))∪U(V(K)),(4)
L(V(H)∩V(K))=L(V(H))∩L(V(K)),(5)
ifH⊆K,thenL(V(H))⊆L(V(K)),(6)
ifH⊆K,thenU(V(H))⊆U(V(K)),(7)
L(V(H)∪V(K))⊇L(V(H))∪L(V(K)),(8)
U(V(H)∩V(K))⊆U(V(H))∩U(V(K)),(9)
L(V(G)-V(H))=V(G)-U(V(H)),(10)
U(V(G)-V(H))=V(G)-L(V(H)).Proof.
By using properties ofGm-interior and Gm-closure, the proof is obvious.The following example illustrates that properties 11 and 12 which are introduced in Section2.1 cannot be applied for this new generalization.Example 3.5.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-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)},ℱG={V(G),ϕ,{v1},{v1,v3},{v1,v2,v4}},𝒯G={V(G),ϕ,{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
(3.2)L(L(V(H)))=L(V(H))={v3},U(L(V(H)))={v1,v3}.
Thus,
(3.3)L(L(V(H)))=L(V(H))≠U(L(V(H))).
Also,
(3.4)U(U(V(H)))=U(V(H))={v1,v2,v4},L(U(V(H)))={v2,v4}.
Thus,
(3.5)U(U(V(H)))=U(V(H))≠L(U(V(H))).Lemma 3.6.
Let (G,ℱGm) be a Gm-closure space. Then
(3.6)IntGm(V(G)-V(H))=V(G)-ClGm(V(H))∀subgraphH⊆G.Proof.
It follows from definition ofGm-closure space.Lemma 3.7.
LetH be a subgraph of G in the Gm-closure space (G,ℱGm). Then v∈ClGm(V(H)) if and only if for each subgraph K⊆G and v∈IntGm(V(K)), then IntGm(V(K))∩V(H)≠ϕ.Proof.
(⇒) Let v∈ClGm(V(H)) and v∈IntGm(V(K)) for some K⊆G. Assume IntGm(V(K))∩V(H)=ϕ. This implies that V(H)⊆V(G)-IntGm(V(K)) which is closed graph. Hence, v∈V(G)-IntGm(V(K)), since v∈ClGm(V(H)) and this leads to a contradiction. Therefore, IntGm(V(K))∩V(H)≠ϕ.
(⇐) Suppose that for each K⊆G and v∈IntGm(V(K)), IntGm(V(K))∩V(H)≠ϕ. Let v∉ClGm(V(H)) which is closed. Then there exists a closed graph F⊆Gsuch that F⊇H and  v∉V(F). Hence, V(G)-V(F) is open subgraph containing v. Thus, v∈IntGm(V(G)-V(F))=V(G)-V(F) and IntGm(V(G)-V(F))∩V(H)=ϕ, that is, there exists a subgraph K=G-F of G such that IntGm(V(K))∩V(H)=ϕ, which leads to a contradiction. Therefore,  v∈ClGm(V(H).Lemma 3.8.
LetH and K be two subgraphs of G in the Gm-closure space (G,ℱGm). If H is open subgraph, then  V(H)∩ClGm(V(K))⊆ClGm(V(H)∩V(K)).Proof.
Letv∈V(H)∩ClGm(V(K)). If O is open subgraph such that v∈V(O), then V(O)∩V(H) is an open subgraph and  v∈V(O)∩V(H). Therefore,  V(O)∩(V(H)∩V(K))≠ϕandv∈ClGm(V(H)∩V(K)). Hence, the result.Proposition 3.9.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H and K are subgraphs of G, then(a)
L(V(H)-V(K))⊆L(V(H))-L(V(K)),(b)
U(V(H)-V(K))⊇U(V(H)-U(V(K)).Proof.
(a) We need to show thatIntGm(V(H)-V(K))⊆IntGm(V(H))-IntGm(V(K)). Now,
(3.7)V(H)-V(K)=V(H)∩(V(G)-V(K)).
Then,
(3.8)IntGm(V(H)-V(K))=IntGm(V(H)∩(V(G)-V(K)))=IntGm(V(H))∩IntGm(V(G)-V(K)).
Thus, by Lemma 3.6, we have
(3.9)IntGm(V(H)-V(K))=IntGm(V(H))∩(V(G)-ClGm(V(K)))=IntGm(V(H))-ClGm(V(K))⊆IntGm(V(H))-IntGm(V(K)).
Therefore,
(3.10)L(V(H)-V(K))=IntGm(V(H)-V(K))⊆IntGm(V(H))-IntGm(V(K))=L(V(H))-L(V(K)).(b) We need to show that
(3.11)ClGm(V(H)-V(K))⊇ClGm(V(H))-ClGm(V(K)).
Now,
(3.12)ClGm(V(H))-ClGm(V(K))=ClGm(V(H))∩(V(G)-ClGm(V(K))).
Thus, by Lemma 3.6, we have
(3.13)ClGm(V(H))-ClGm(V(K))=ClGm(V(H))∩IntGm(V(G)-V(K)).
Hence, by Lemma 3.8, we have
(3.14)ClGm(V(H))-ClGm(V(K))=ClGm(V(H))∩IntGm(V(G)-V(K))⊆ClGm[V(H)∩IntGm(V(G)-V(K))]=ClGm[V(H)∩V(G)-ClGm(V(K))]=ClGm[V(H)-ClGm(V(K))],
Thus,
(3.15)ClGm(V(H))-ClGm(V(K))⊆ClGm(V(H)-V(K)).
Therefore,
(3.16)U(V(H)-V(K))=ClGm(V(H)-V(K))⊇ClGm(V(H))-ClGm(V(K))=U(V(H))-U(V(K)).


## 4. Near Lower and Near Upper inGm-Closure Approximation Spaces

In this section, we study approximation spaces fromGm-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.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space and H⊆G. The near-lower approximation (j-lower approximation) (resp., near-upper approximation (j-upper approximation)) of H is denoted by Lj(V(H))(resp.,Uj(V(H))) and is defined by
(4.1)Lj(V(H))=IntGmj(V(H))(resp.,Uj(V(H))=ClGmj(V(H))),wherej∈{R,S,P,γ,α,β}.Proposition 4.2.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If  H⊆G, then L(V(H))⊆Lj(V(H))⊆V(H)⊆Uj(V(H))⊆U(V(H)), for all j∈{S,P,γ,α,β}.Proof.
The proofs of the five cases are similar, so we will only prove the case whenj=S. Now,
(4.2)U(V(H))=ClGm(V(H))=∩{V(F);V(F)∈FGmandV(H)⊆V(F)}⊇∩{V(F);V(F)∈SCGm(G)andV(H)⊆V(F)}sinceFGm⊆SCGm(G)=ClGmS(V(H))=US(V(H))⊇V(H),(4.3)L(V(H))=IntGm(V(H))=V(G)-ClGm(V(G)-V(H))⊆V(G)-ClGmS(V(G)-V(H))sinceTGm⊆SOGm(G)=IntGmS(V(H))=LS(V(H))⊆V(H).
From (4.2) and (4.3) we get L(V(H))⊆LS(V(H))⊆V(H)⊆US(V(H))⊆U(V(H)).In general the above proposition is not true in the case ofj=R as the following example illustrates.Example 4.3.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space such that G=(V(G),E(G)): V(G)={v1,v2,v3},E(G)={(v2,v1),(v2,v3)},ℱG={V(G),ϕ,{v1},{v3},{v1,v3}},𝒯G={V(G),ϕ,{v2},{v1,v2},{v2,v3}}.Hence, ROGm(G)={V(G),ϕ}andRCGm(G)={V(G),ϕ}. If H=(V(H),E(H)): V(H)={v1,v3},E(H)=ϕ, then
(4.4)L(V(H))=IntGm(V(H))=ϕ,U(V(H))=ClGm(V(H))={v1,v3},LR(V(H))=IntGmR(V(H))=ϕ,UR(V(H))=ClGmR(V(H))=V(G).
Therefore,
(4.5)LR(V(H))=L(V(H)),U(V(H))⊆UR(V(H)).Proposition 4.4.
Let𝒢m=(G,ClGm,ℱGm) be a Gm- closure approximation space. If H⊆G, then the implication between lower approximation and j-lower approximation of H are given by the following statement for all j∈{S,P,γ,α,β}:(a)
L(V(H))⊆Lα(V(H))⊆LS(V(H))⊆Lγ(V(H))⊆Lβ(V(H)),(b)
L(V(H))⊆Lα(V(H))⊆LP(V(H))⊆Lγ(V(H))⊆Lβ(V(H)).Proof.
By using Proposition4.2, we get L(V(H))⊆Lα(V(H)). We will prove Lα(V(H))⊆LS(V(H)). Now,
(4.6)Lα(V(H))=IntGmα(V(H))=V(G)-ClGmα(V(G)-V(H))⊆V(G)-ClGmS(V(G)-V(H)),
since αOGm(G)⊆SOGm(G). Thus,
(4.7)Lα(V(H))=IntGmα(V(H))⊆IntGmS(V(H))=LS(V(H)).
Similarly we can prove the other cases.Proposition 4.5.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H⊆G, then the implication between upper approximation and j-upper approximation of H are given by the following statement for all j∈{S,P,γ,α,β},(a)
Uβ(V(H))⊆Uγ(V(H))⊆US(V(H))⊆Uα(V(H))⊆U(V(H)),(b)
Uβ(V(H))⊆Uγ(V(H))⊆UP(V(H))⊆Uα(V(H))⊆U(V(H)).Proof.
By using Proposition4.2, we get Uα(V(H))⊆U(V(H)). We will prove UP(V(H))⊆Uα(V(H)). Now,
(4.8)UP(V(H))=ClGmP(V(H))=∩{V(F);V(F)∈PCGm(G)andV(H)⊆V(F)}⊆∩{V(F);V(F)∈αCGm(G)andV(H)⊆V(F)}
since αCGm(G)⊆PCGm(G). Thus,
(4.9)UP(V(H))=ClGmP(V(H))⊆ClGmα(V(H))=Uα(V(H)).
Similarly we can prove the other cases.Proposition 4.6.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H and K are two subgraphs of G, then, for all j∈{R,S,P,γ,α,β},(1)
Lj(ϕ)=Uj(ϕ)=ϕandLj(V(G))=Uj(V(G))=V(G),(2)
ifV(H)⊆V(K),thenLj(V(H))⊆Lj(V(K)),(3)
ifV(H)⊆V(K),thenUj(V(H))⊆Uj(V(K)),(4)
Lj(V(H)∪V(K))⊇Lj(V(H))∪Lj(V(K)),(5)
Uj(V(H)∪V(K))⊇Uj(V(H))∪Uj(V(K)),(6)
Lj(V(H)∩V(K))⊆Lj(V(H))∩Lj(V(K)),(7)
Uj(V(H)∩V(K))⊆Uj(V(H))∩Uj(V(K)),(8)
Lj(V(H)c)=[Uj(V(H))]c,(9)
Uj(V(H)c)=[Lj(V(H))]c.Proof.
By using properties ofj-interior and j-closure for all  j∈{R,S,P,γ,α,β}, the proof is obvious.In general, properties 3 and 4 which are introduced in Section2.1 cannot be applied for j-lower and j-upper approximations, where j∈{S,P,γ,β}. The following example illustrates this fact in the case of j=β.Example 4.7.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space which is given in Example 2.3:ℱG={V(G),ϕ,{v3},{v3,v4},{v1,v2,v3}},𝒯G={V(G),ϕ,{v4},{v1,v2},{v1,v2,v4}}.(4.10)βOG1(G)={V(G),ϕ,{v1},{v2},{v4},{v1,v2},{v1,v3}{v1,v4},{v2,v3},{v2,v4},{v3,v4},{v1,v2,v3},{v1,v2,v4},{v1,v3,v4},{v2,v3,v4}},βCG1(G)={V(G),ϕ,{v1},{v2},{v3},{v4},{v1,v2}{v1,v3},{v1,v4},{v2,v3},{v2,v4},{v3,v4},{v1,v2,v3},{v1,v3,v4},{v2,v3,v4}}.
If
(4.11)H=(V(H),E(H));V(H)={v1,v3},E(H)={(v1,v3)},K=(V(K),E(K));V(K)={v2,v3},E(K)={(v2,v3)},
then
(4.12)Lβ(V(H))∩Lβ(V(K))={v1,v3}∩{v2,v3}={v3},
but
(4.13)Lβ(V(H)∩V(K))=ϕ.
Thus,
(4.14)Lβ(V(H)∩V(K))≠Lβ(V(H))∩Lβ(V(K)).
Also, if
(4.15)H=(V(H),E(H));V(H)={v1,v2},E(H)={(v1,v2),{v2,v1}},K=(V(K),E(K));V(K)={v1,v4},E(K)=ϕ,
then
(4.16)Uβ(V(H))∪Uβ(V(K))={v1,v2}∪{v1,v4}={v1,v2,v4},
but
(4.17)Uβ(V(H)∪V(K))=V(G).
Thus,
(4.18)Uβ(V(H)∪V(K))≠Uβ(V(H))∪Lβ(V(K)).In general, properties 11 and 12 which are introduced in Section2.1 cannot be applied for j-lower and j-upper approximations, where j∈{S,P,γ,β}. The following example illustrates this fact in the case of  j=β.Example 4.8.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space which is given in Example 2.3. If
(4.19)H=(V(H),E(H));V(H)={v1,v2,v4},E(H)={(v1,v2),(v2,v1)},K=(V(K),E(K));V(K)={v3},E(K)=ϕ,
then
(4.20)Lβ(Lβ(V(H)))=Lβ(V(H))={v1,v2,v4},Uβ(Lβ(V(H)))=V(G).
Thus,
(4.21)Lβ(Lβ(V(H)))=Lβ(V(H))≠Uβ(Lβ(V(H))).
Also,
(4.22)Uβ(Uβ(V(K)))=Uβ(V(K))={v3},Lβ(Uβ(V(K)))=ϕ.
Hence,
(4.23)Uβ(Uβ(V(K)))=Uβ(V(K))≠Lβ(Uβ(V(K))).Lemma 4.9.
Let(G,ℱGm) be a Gm-closure space. Then  IntGmj(V(G)-V(H))=V(G)-ClGmj(V(H)) for all subgraph H⊆G  and j∈{R,S,P,γ,α,β}.Proof.
It follows from definition near-open subgraphs inGm-closure space.Proposition 4.10.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H and K are subgraphs of G, then
(4.24)Lj(V(H)-V(K))⊆Lj(V(H))-Lj(V(K)),∀j∈{R,S,P,γ,α,β}.Proof.
We need to show that(4.25)IntGmj(V(H)-V(K))⊆IntGmj(V(H))-IntGmj(V(K)).
Now,
(4.26)V(H)-V(K)=V(H)∩(V(G)-V(K)).
Then
(4.27)IntGmj(V(H)-V(K))=IntGmj(V(H)∩(V(G)-V(K)))⊆IntGmj(V(H))∩IntGmj(V(G)-V(K)).
Thus, by Lemma 4.9, we have
(4.28)IntGmj(V(H)-V(K))⊆IntGmj(V(H))∩(V(G)-ClGmj(V(K)))=IntGmj(V(H))-ClGmj(V(K))⊆IntGmj(V(H))-IntGmj(V(K)).
Therefore
(4.29)Lj(V(H)-V(K))=IntGmj(V(H)-V(K))⊆IntGmj(V(H))-IntGmj(V(K))=Lj(V(H))-Lj(V(K)).In general, part (b) in Proposition3.9 cannot be applied for j-upper approximations for all j∈{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.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space which is given in Example 2.3. If
(4.30)H=(V(H),E(H));V(H)={v1,v2,v4},E(H)={(v1,v2),(v2,v1)},K=(V(K),E(K));V(K)={v1,v2},E(K)={(v1,v2),(v2,v1)},
then
(4.31)Uβ(V(H)-V(K))=Uβ({v4})={v4},
but
(4.32)Uβ(V(H))-Uβ(V(K))=V(G)-{v1,v2}={v3,v4}.
Hence,
(4.33)Uβ(V(H)-V(K))⊆Uβ(V(H))-Uβ(V(K)).Example 4.12.
Let𝒢m=(G,ClGm,ℱGm)  be a Gm-closure approximation space which is given in Example 2.3:
(4.34)ROG1(G)={V(G),ϕ,{v4},{v1,v2}},RCG1(G)={V(G),ϕ,{v3,v4},{v1,v2,v3}}.
If
(4.35)H=(V(H),E(H));V(H)={v1},E(H)=ϕ,K=(V(K),E(K));V(K)={v3},E(K)=ϕ,
then
(4.36)UR(V(H)-V(K))=UR(ϕ)=ϕ,
but
(4.37)UR(V(H))-UR(V(K))={v1,v2,v3}-{v3}={v1,v2}.
Hence,
(4.38)UR(V(H)-V(K))⊆UR(V(H))-UR(V(K)).


## 5. Near-Boundary Regions and Near Accuracy inGm-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 subgraphH of G in a Gm-closure approximation space 𝒢m=(G,ClGm,ℱGm).Definition 5.1.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space and  H⊆G. The near-boundary (j-boundary) region of H is denoted by Bd𝒢mj(V(H)) and is defined by
(5.1)BdGmj(V(H))=Uj(V(H))-Lj(V(H)),wherej∈{R,S,P,γ,α,β}.Definition 5.2.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space and H⊆G. The near-positive (j-positive) region of H is denoted by POS𝒢mj(V(H)) and is defined by
(5.2)POSGmj(V(H))=Lj(V(H)),wherej∈{R,S,P,γ,α,β}.Definition 5.3.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space and  H⊆G. The near negative (briefly j-negative) region of H is denoted by NEG𝒢mj(V(H)) and is defined by
(5.3)NEGGmj(V(H))=V(G)-Uj(V(H)),wherej∈{R,S,P,γ,α,β}.Proposition 5.4.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H⊆G, then
(5.4)BdGmj(V(H))⊆BdGm(V(H))∀j∈{S,P,γ,α,β}.Proof.
By using Proposition4.2, the proof is obvious.In general, the above proposition is not true in the case ofj=R as illustrated in the following example.Example 5.5.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space which is given in Example 2.3. If
(5.5)H=(V(H),E(H)):V(H)={v1,v3},E(H)={(v1,v3)},
then
(5.6)BdGm(V(H))=U(V(H))-L(V(H))=ClGm(V(H))-IntGm(V(H))=ClGm({v1,v3})-IntGm({v1,v3})={v1,v3}-ϕ={v1,v3},BdGmR(V(H))=UR(V(H))-LR(V(H))=ClGmR(V(H))-IntGmR(V(H))=ClGmR({v1,v3})-IntGmR({v1,v3})=V(G)-ϕ=V(G).Proposition 5.6.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H⊆G, then the implication between boundary and j-boundary of H given by the following statement for all j∈{S,P,γ,α,β}:(a)
Bd𝒢mβ(V(H))⊆Bd𝒢mγ(V(H))⊆Bd𝒢mS(V(H))⊆Bd𝒢mα(V(H))⊆Bd𝒢m(V(H)),(b)
Bd𝒢mβ(V(H))⊆Bd𝒢mγ(V(H))⊆Bd𝒢mP(V(H))⊆Bd𝒢mα(V(H))⊆Bd𝒢m(V(H)).Proof.
By using Propositions4.4 and 4.5, the proof is obvious.Definition 5.7.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space and H a finite nonempty subgraph of G. The near accuracy (j-accuracy) of H is denoted by η𝒢mj(V(H)) and is defined by
(5.7)ηGmj(V(H))=|Lj(V(H))||Uj(V(H))|,where|Uj(V(H))|≠0∀j∈{R,S,P,γ,α,β}.Proposition 5.8.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H is a finite nonempty subgraph of G, then  η𝒢m(V(H))≤η𝒢mj(V(H)) for all j∈{S,P,γ,α,β}, where η𝒢m(V(H))=|L(V(H))|/|U(V(H))|is the accuracy of H.Proof.
By using Proposition4.2, the proof is obvious.In general, the above proposition is not true in the case ofj=R. This fact is illustrated in the following example.Example 5.9.
Let𝒢m=(G,ClGm,ℱGm)  be a Gm-closure approximation space which is given in Example 4.3. If
(5.8)H=(V(H),E(H)):V(H)={v1,v2},E(H)={(v2,v1)},
then
(5.9)ηGm(V(H))=23,ηGmR(V(H))=0.
Thus,
(5.10)ηGmR(V(H))<ηGm(V(H)).Proposition 5.10.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If H⊆G, then the implication between accuracy and j-accuracy of H is given by the following statement for all j∈{S,P,γ,α,β}:(a)
η𝒢m(V(H))≤η𝒢mα(V(H))≤η𝒢mS(V(H))≤η𝒢mγ(V(H))≤η𝒢mβ(V(H)),(b)
η𝒢m(V(H))≤η𝒢mα(V(H))≤η𝒢mP(V(H))≤η𝒢mγ(V(H))≤η𝒢mβ(V(H)).Proof.
By using Propositions4.4 and 4.5, the proof is obvious.


## 6. Rough and Near-Rough Cluster Vertices inGm-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 subgraphH⊆G in a Gm-closure approximation space 𝒢m=(G,ClGm,ℱGm).Definition 6.1.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. If  H⊆G, then H is called(a)
totallyj𝒢m-definable (j𝒢m-exact) graph if Lj(V(H))=V(H)=Uj(V(H)),(b)
internallyj𝒢m-definable graph if Lj(V(H))=V(H),Uj(V(H))≠V(H),(c)
externallyj𝒢m-definable graph if Lj(V(H))≠V(H),Uj(V(H))=V(H),(d)
j𝒢m-indefinable (j𝒢m-rough) graph if  Lj(V(H))≠V(H),Uj(V(H))≠V(H), where j∈{R,S,P,γ,α,β}.Example 6.2.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-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,v2)}, ℱG={V(G),ϕ,{v1},{v1,v3},{v1,v2,v4}},𝒯G={V(G),ϕ,{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 j∈{S,P}, we get
(6.1)POSGmS(V(H))=LS(V(H))=IntGmS(V(H))={v1,v3},US(V(H))=ClGmS(V(H))=V(G),BdGmS(V(H))=BdGmS(V(H))={v2,v4},NEGGmS(V(H))=ϕ,POSGmP(V(H))=LP(V(H))=IntGmP(V(H))={v1,v2,v3},UP(V(H))=ClGmP(V(H))={v1,v2,v3},BdGmS(V(H))=BdGmP(V(H))=ϕ,NEGGmP(V(H))={v4}.
Thus, H is an S𝒢m-indefinable (S𝒢m-rough) graph and P𝒢m-definable (P𝒢m-exact) graph.
The following definition introduces the concept of rough cluster vertices of a subgraphH of G in a Gm-closure approximation 𝒢m=(G,ClGm,ℱGm).Definition 6.3.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. The vertex v∈G is said to be a rough cluster vertex of a subgraph H of G if, for all subgraph K of G such that v∈L(V(K)),  (L(V(K))-{v})∩V(H)≠ϕ.
The graph of all rough cluster vertices ofH is denoted by R′(V(H)) and is called the rough derived graph of H.Theorem 6.4.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. Then a subgraph H of G is closed if and only if R′(V(H))⊆V(H).Proof.
(⇒) Suppose that H is a closed subgraph of G, and let v∉V(H) (i.e., v∈V(G)-V(H)). Then V(G)-V(H) is open subgraph. Thus,  v∈L(V(G)-V(H))=IntGm(V(G)-V(H))=V(G)-V(H) and L(V(G)-V(H))∩V(H)=ϕ. Hence, v∉R′(V(H)). Therefore, R′(V(H))⊆V(H).
(⇐) Let R′(V(H))⊆V(H). To show that H is a closed subgraph of  G, let v∈V(G)-V(H). Then v∉R′(V(H)), and hence there exists a subgraph Kv⊆G such that v∈L(V(Kv)) and L(V(Kv)-V(v))∩V(H)=ϕ. But v∉V(H), hence L(V(Kv))∩V(H)=ϕ. So v∈L(V(Kv)⊆V(G)-V(H) and V(G)-V(H)=⋃v∈V(G)-V(H){v}⊆⋃v∈V(G)-V(H)L(V(Kv)⊆⋃v∈V(G)-V(H)IntGm(V(Kv))⊆V(G)-V(H).
Thus,V(G)-V(H) is a union of open graphs, which is open. Hence,  H is closed subgraph of G.Example 6.5.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space such that G=(V(G),E(G)): V(G)={v1,v2,v3,v4}, E(G)={(v1,v2),(v1,v3),(v2,v1),(v2,v3),(v4,v3)}, ℱG={V(G),ϕ,{v3},{v3,v4},{v1,v2,v3}},𝒯G={V(G),ϕ,{v4},{v1,v2},{v1,v2,v4}}.If H=(V(H),E(H)): V(H)={v1,v2,v3}, E(H)={(v1,v2),(v1,v3),(v2,v1),(v2,v3)}, then R′(V(H))={v1,v2,v3}. Thus, R′(V(H))⊆V(H) and H is closed subgraph of G.
The following definition introduces the concept of near-rough (j-rough) cluster vertices of a subgraph H of G in a Gm-closure approximation space 𝒢m=(G,ClGm,ℱGm) for all j∈{R,S,P,γ,α,β}.Definition 6.6.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. The vertex v∈G is said to be near-rough (j-rough) cluster vertex of a subgraph H of G for all j∈{R,S,P,γ,α,β}, if, for all subgraph K of G such that v∈Lj(V(K)), (Lj(V(K))-{v})∩V(H)≠ϕ.
The graph of allj-rough cluster vertices of H is denoted by Rj′(V(H))and is called the j-rough derived graph of H for all j∈{R,S,P,γ,α,β}.Theorem 6.7.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space. Then a subgraph H of  G is a j-closed for all j∈{S,P,γ,α,β} if and only if Rj′(V(H))⊆V(H).Proof.
The proofs of the five cases are similar, so we will only prove the case whenj=β.
(⇒) Suppose that H is a β-closed subgraph of G, and let v∉V(H) (i.e., v∈(V(G)-V(H)). Then V(G)-V(H)∈βOGm(G). Thus, v∈Lβ(V(G)-V(H))=IntGmβ(V(G)-V(H))=V(G)-V(H) and Lβ(V(G)-V(H))∩V(H)=ϕ. Hence, v∉Rβ′(V(H)). Therefore, Rβ′(V(H))⊆V(H).
(⇐) Let Rβ′(V(H))⊆V(H). To show that H is a β-closed subgraph of G, let v∈(V(G)-V(H)), then v∉R′(V(H)), and hence there exists a subgraph Kv⊆G such that v∈Lβ(V(Kv)) and Lβ(V(Kv)-V(v))∩V(H)=ϕ. But v∉V(H), hence Lβ(V(Kv))∩V(H)=ϕ. So v∈Lβ(V(Kv)⊆V(G)-V(H) and V(G)-V(H)=⋃v∈V(G)-V(H){v}⊆⋃v∈V(G)-V(H)Lβ(V(Kv)⊆⋃v∈V(G)-V(H)IntGmβ(V(Kv))⊆V(G)-V(H).
Thus,V(G)-V(H) is a union of β-open graphs, which is β-open. Hence, H is β-closed subgraph of G.Example 6.8.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space which is given in Example 6.5.
IfH=(V(H),E(H));V(H)={v1,v2},E(H)={(v1,v2),(v2,v1)}. Then RS′(V(H))={v1,v2},thus RS′(V(H))⊆V(H) and H is S-closed subgraph of G.
In general, Theorem6.7 cannot be satisfied in the case of j=R, as the following example illustrates.Example 6.9.
Let𝒢m=(G,ClGm,ℱGm) be a Gm-closure approximation space which is given in Example 6.5.
IfH=(V(H),E(H));V(H)={v3},E(H)=ϕ. Then RR′(V(H))={v3}, thus RR′(V(H))⊆V(H). But H is not an R-closed subgraph of  G, since RCGm(G)={V(G),ϕ,{v3,v4},{v1,v2,v3}}.Theorem 6.10.
LetH be a subgraph of G in the Gm-closure approximation space 𝒢m=(G,ClGm,ℱGm). Then v∈U(V(H)) if and only if, for each K⊆G and v∈L(V(K), L(V(K))∩V(H)≠ϕ.Proof.
(⇒) Let v∈U(V(H)) and v∈L(V(K)) for some K⊆G. Assume L(V(K))∩V(H)=ϕ. This implies that V(G)⊆V(G)-L(V(K)). But V(G)-L(V(K))=V(G)-IntGm(V(K)) which is closed graph. Hence, v∈V(G)-L(V(K)), since v∈U(V(H)) and this leads to a contradiction. Therefore, L(V(K))∩V(H)≠ϕ.
(⇐) Suppose that, for each K⊆G and v∈L(V(K)),  L(V(K))∩V(H)≠ϕ. Let v∉U(V(H)). But U(V(H))=ClGm(V(H)) which is closed. Then there exists a closed graph F⊆G such that F⊇H and v∉V(F). Hence, V(G)-V(F) is open graph containing v. Thus,
(6.2)v∈L(V(G)-V(F))=IntGm(V(G)-V(F))=V(G)-V(F),L(V(G)-V(F))∩V(H)=ϕ.
that is, there exists a subgraph K=G-F such that L(V(K)∩V(H)=ϕ, which leads to a contradiction. Therefore, v∈U(V(H)).Theorem 6.11.
LetH be a subgraph of G in the Gm-closure approximation space 𝒢m=(G,ClGm,ℱGm). Then v∈Uj(V(H)) for all j∈{S,P,γ,α,β} if and only if, for each K⊆G and v∈Lj(V(K)), L(V(K))∩V(H)≠ϕ.Proof.
The proof is similar to the proof of Theorem6.10.Theorem 6.12.
LetH be a subgraph of G in the Gm-closure approximation space  𝒢m=(G,ClGm,ℱGm). Then  U(V(H)=V(H)∪R′(V(H)).Proof.
By Theorem6.4, we get R′(V(H))⊆U(V(H)). Then
(6.3)V(H)∪R′(V(H))⊆V(H)∪U(V(H))=U(V(H)).
For the converse inclusion, let v∈U(V(H)), then either v∈V(H) and hence v∈V(H)∪R’(V(H)) or v∉V(H). Hence, by Theorem 6.10 for each K⊆G, v∈L(V(K)), we get L(V(K))∩V(H)≠ϕ. Then v∈R′(V(H)) and hence v∈V(H)∪Rj′(V(H)). Thus, U(V(H))⊆V(H)∪Rj′(V(H)). Therefore, U(V(H)=V(H)∪Rj′(V(H)).Theorem 6.13.
LetH be a subgraph of G in the Gm-closure approximation space 𝒢m=(G,ClGm,ℱGm). Then Uj(V(H))=V(H)⋃Rj′(V(H))  for all j∈{S,P,γ,α,β}.Proof.
The proof is similar to the proof of Theorem6.12.


## 7. Conclusions

In this paper, we usedGm-topological concepts to introduce a generalization of Pawlak approximation space. Concepts of definability for subgraphs in Gm-approximation spaces are introduced. Several types of approximations which are called near approximations are mathematical tools to modify the approximations. The suggested methods of near approximations open way for constructing new types of lower and upper approximations.



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