# Different Neighbourhoods via Ideals on Graphs

Authors: A. Çaksu Güler
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we generate different approximations forj∈r,l,r,l,u,i,u,i by using the concept of ideal, Nj-neighbourhoods and Pj-neighbourhoods from vertices of a graph. Then, we compare these approximations with the previous one, and we see that these approximations are more accurate. Finally, we show that our approach is better with a real-life problem from the field of chemistry by using these new approximations.


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## Body

## 1. Introduction

Due to the importance of mathematically expressing uncertain concepts, which can’t be defined by classical logic, researchers offer new theories every day. Some of the most essential and well-known theories are probability and statistics.Researchers in economics, medicine, engineering, and business are struggling with the complexity of uncertain data modeling every day. Classical methods can’t always be successful because of the uncertainties that arise in these areas can take various forms. Rough set theory is a well-known and useful approach to uncertainty identification. It was presented, as a point of mathematical view by Pawlak [1, 2] that interested in the uncertainty and ambiguity of imprecise data. The central idea in this theory is an approximation operator which is identified in equivalence classes. However, such an equivalence relation is insufficient for use in some current applications. Therefore, many generalizations of equivalence relations have been proposed, such as tolerance relations, preorder relations, and arbitrary binary relations [3–5]. It has many varieties of applications in many fields such as biology data analysis, chemistry, and engineering.Lin [6] and Yao [7] investigated rough sets by using neighbourhood systems for the interpretation of granules. Abd El-Monsef et al. [8] defined mixed neighbourhood systems to approximate rough sets. Then, Abd El-Monsef et al. [8] presented neighbourhood systems and the j-neighbourhood space which represents a generalized type of neighbourhood spaces. Amer et al. [9] obtained new j-nearly approximations as mathematical tools for modifying and generalizing the j-approximations in the j-neighbourhood space. Atef et al. [10] gave the generalization of some types of rough set models based upon j-neighbourhood space and other types of rough set models based on j-adhesion neighbourhood space. They also introduced the notions of Pj-neighbourhoods using j-neighbourhoods and investigated their properties. Then, Al-Shami [11] established the notions of Cj-neighbourhoods using j-neighbourhoods and studied their characteristics. Also, Al-Shami and Ciucci [12] studied the notions of Sj-neighbourhoods using j-neighbourhoods and investigated their properties. Lastly, Al-Shami [13] defined and studied a new class of neighbourhoods system which is called Mj-neighbourhoods.Ideals, which play an important role and are fundamental concept in topological spaces, are first studied by Kuratowski [14]. Then, Kandil et al. [15] generalized Pawlak’s approximations by using this concept with r-neighbourhoods. Later, Hosny [16] generated different topologies using the concept of the ideal. Hosny constructed new approximations namely ℐ-j-approximations using these topologies. Also, she proved that these approximations are extended the notation of j-approximations. Hosny [17] presented the concepts of j-nearly open sets with respect to ideals and j-nearly approximations in terms of ideals as generalizations of j-nearly approximations. Then, Hosny et al. [18] produced some topologies by using the concept of ideal and Ej-neighbourhoods [19]. They also studied a new kind of approximation and compared them with the previous one. Recently, Caksu Guler et al [20] studied various topologies generated by ideals, Cj-neighbourhoods and Pj-neighbourhoods. Then, they introduced new methods to find the approximations by using these generated topologies.Graph theory [21, 22] is a very famous field with a wide range of uses, and many of them are in different fields, for example, marketing, business, management, information transmission, mathematics, chemistry, biology, and physics. It allows the problems to be simplified and solved with good, effective, and neat steps. Simply, a graph is a collection of vertices and edges where the edges connect all vertices or just the vertex itself. Mathematically, a graph is one of the sets G=V,E=VG,EG is a pair of vertices sets V and edges sets E. Two vertices are adjacent if there is an edge between two vertices. An isolated vertex is a vertex that is not an end of any edge. Much of graph theory is concerned with the study of simple graphs. A graph G is simple if every edge links a unique pair of distinct vertices. Let G=VG,EG be a graph; we call H a subgraph of G if VH⊂VG and EH⊂EG, and under the circumstances, we write H⊂G. Many studies on graph theory and its applications have been made by researchers. Some of those studies on graph theory are Järvinen [23] where binary relations are represented by a graph, and Chen and Li [24] studied where binary relations are represented by a directed simple graph. From this point of view, the relations between some concepts of graph theory and rough set theory have been started to be examined. By using neighbourhood system on rough sets, Nada et al. [25] in 2018 started to work on topological structure on graphs that represent the right neighbourhoods. Then, El Atik and Nasef [26, 27] used the rough sets on graphs and the neighbourhood systems on different fields used in medicine and physics to represent structures as a human hearth and self-similar fractals. In 2021, El Atik et al. [28] studied neighborhood system which is based on the vertices of a graph is defined, and properties are investigated.The layout of this paper is as follows: in Section2, we have included the definitions and theorems that we will be used throughout the paper. In Section 3, we define a new class of approximations called ℐ-N-j-approximations, where j∈r,l,r,l,u,i,u,i by using ideal and j-neighbourhoods for any subgraph of a given graph. Then, we study the relationships between them. Besides, we formulate the concepts of ℐ-N-j-boundary region and ℐ-N-j-accuracy of approximations of a subgraph. We compare them with those given by Al-Atik [28] and obtain that they are more accurate. In Section 4, we gave the definition ℐ-P-j-lower and ℐ-P-j-upper approximations, ℐ-P-j-boundary region, and ℐ-P-j-accuracy measure of a subgraph VH of VG by using ideals and Pj-neighbourhoods. Moreover, we show the relationship between these notions by giving some examples and investigated their properties. Also, we compare these approximations with the ones that we give in the previous section. We summarize all comparisons that we made throughout the paper with tables and we give counterexamples for supporting the study. Then, we give a simple example which we apply these results on.


## 2. Preliminaries

In this section, we gave concepts such as rough sets, graph theory, and neighbourhood.Definition 1 (see [14]).
LetX be a nonempty set. Then, a family of sets ℐ⊂PX is said to be an ideal in X if(a)
A,B∈ℐ imply A∪B∈ℐ(b)
A∈ℐ and B⊂A imply B∈ℐDefinition 2 (see [7]).
LetR be a binary relation on a nonempty set U and X⊆U. Lower approximations and under approximations of X are defined by(a)
R¯X=x∈U:xR⊆X(b)
R¯X=x∈U:xR∩A≠∅, where xR=y∈U:xRyProposition 1 (see [1, 2]).
LetR be a binary relation on a nonempty set U and X,Y⊆U. Then, the following properties hold:L1R¯X⊆XL2R¯∅=∅L3R¯U=UL4R¯X∩Y=R¯X∩R¯YL5IfX⊆Y,thenR¯X⊆R¯YL6R¯X∪R¯Y⊆R¯X∪YL7R¯Xc=(R¯X)cL8R¯R¯X=R¯XL9R¯R¯Xc=(R¯X)cU1X⊆R¯XU2R¯∅=∅U3R¯U=UU4R¯X∪Y=R¯X∪R¯YU5IfX⊆Y,thenR¯X⊆R¯YU6R¯X∩R¯Y⊇R¯X∪YU7R¯Xc=(R¯X)cU8R¯R¯X=R¯XU9R¯R¯Xc=(R¯X)cDefinition 3 (see [7, 28, 29]).
LetG=VG,EG be a graph for each v∈VG. The j-neighbourhood of v, j∈J=r,l,r,l,u,i,u,i is defined as(a)
Nrv=w∈VG:vRw(b)
Nlv=w∈VG:wRv(c)
Nrv=∩v∈NrwNrw∃Nrwcontainingv∅Otherwise(d)
Nlv=∩v∈NlwNlw∃Nlwcontainingv∅Otherwise(e)
Nuv=Nrv∪Nlv(f)
Niv=Nrv∩Nlv(g)
Nuv=Nrv∪Nlv(h)
Niv=Nrv∩NlvDefinition 4 (see [28]).
LetG be a graph and H be a subgraph of G and j∈J. The first-type lower j-approximations and upper j-approximations are defined, respectively, by(a)
N¯jVH=v∈VG:Njv⊆VH(b)
N¯jVH=VH∪v∈VG:Njv∩VH≠∅Definition 5 (see [28]).
LetG be a graph and H be a subgraph of G and j∈J. j-boundary regions and j-accuracy measures of H in terms of j-adhesion neighbourhood are defined, respectively, by(a)
BNjVH=N¯jVH−N¯jVH(b)
σNjVH=N¯jVH/N¯jVH, where N¯jVH≠0Definition 6 (see [10]).
LetG be a graph for each v∈VG. Then, the Pj-neighbourhoods of v, j∈J are defined as(a)
Prv=w∈VG:Nrw=Nrv(b)
Plv=w∈VG:Nlw=Nlv(c)
Puv=Prv∪Plv(d)
Piv=Prv∩Plv(e)
Prv=w∈VG:Nrv=Nrw(f)
Plv=w∈VG:Nlv=Nlw(g)
Puv=Prv∪Plw(h)
Piv=Prv∩PlwDefinition 7 (see [28]).
LetG be a graph and H be a subgraph of G and j∈J. The first type lower j-approximations and upper j-approximations are defined, respectively, by(a)
P¯jVH=v∈VG:Pjv⊆VH(b)
P¯jVH=VH∪v∈VG:Pjv∩VH≠∅Definition 8 (see [28]).
LetG be a graph and H be a subgraph of G and j∈J. j-boundary regions and j-accuracy measure of H in terms of j-adhesion neighbourhood are defined, respectively, by(a)
BPDjVH=P¯jVH−P¯jVH(b)
σPjVH=P¯jVH/P¯jVH, where P¯jVH≠0


## 3.I-N-J Approximation

In this section, the concepts ofℑNj-lower and ℑNj-upper approximations are given by using Nj-neighbourhoods for j∈J and ideal. Their essential properties and relationships among them are also examined. Besides, the accuracy measure of a subgraph is defined by using them. Also, it is shown that ℑNj-accuracy measures are more precise when we compare them with Nj-accuracy measures for j∈J.Definition 9.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, the ℑNj-lower and ℑNj-upper approximations of H are defined, respectively, by(a)
N¯jℑVH=v∈VG:Njv∩VH′∈ℑ(b)
N¯jℑVH=VH∪v∈VG:Njv∩VH∈ℑTheorem 1.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the followings hold:(a)
N¯jVH⊆N¯jℑVH(b)
N¯jℑVH⊆N¯jVHProof.
(a)
Letv∈N¯jVH. Then, Njv⊆VH. Therefore, we obtain Njv∩VH′∈ℑ. Hence, v∈N¯jℑVH(b)
The proof is similar to (a).Remark 1.
Ifℑ=∅ in Definition 9, N¯jℑVH (N¯jℑVH) coincides with N¯jVHN¯jVH for each j∈J.Example 1.
LetG be a graph as shown in Figure 1 and ℑ=∅,v1.(a)
LetVM=v2,v3. Then, N¯lVM=v5 and N¯lℑVM=v3,v4,v5. So N¯lVM≠N¯lℑVM.(b)
LetVM=v1,v3. Then, N¯lVM=v1,v2,v3,v4,N¯lℑVM=v1,v2,v3. So N¯lℑVM≠N¯lVM.Figure 1
A simple graphG.Proposition 2.
LetG be a graph, M and N be subgraphs of G and ℑ be an ideal on G. Then, for every j∈J, the followings hold:(a)
VM⊆N¯jℑVM(b)
VM⊆VN implies N¯jℑVM⊆N¯jℑVN; VM⊆VN implies N¯jℑVM⊆N¯jℑVN(c)
N¯jℑVG=N¯jℑVG=VG, N¯jℑ∅=∅(d)
N¯jℑVM∪VN⊇N¯jℑVM∪N¯jℑVN; N¯jℑVM∪VN=N¯jℑVM∪N¯jℑVN(e)
N¯jℑVM∩VN=N¯jℑVM∩N¯jℑVN; N¯jℑVM∩VN⊆N¯jℑVM∩N¯jℑVN(f)
IfVM′∈ℑ, then N¯jℑVM=VG; if VM∈ℑ, then N¯jℑVM=VMProof.
(b) Letv∈N¯jℑVM. Then, Njv∩VM′∈ℑ. Since VM⊆VN and by the definition of ideal, Njv∩VN′∈ℑ. So N¯jℑVM⊆N¯jℑVN.(d) We obtain thatN¯jℑVM∪VN⊇N¯jℑVM∪N¯jℑVN by (b). Let v∈N¯jℑVM∪VN. Then, v∈VM∪VN or v∈w∈VG:Njw∩VM∪VN∈ℑ. If v∈VM∪VN, then we have v∈N¯jℑVM∪N¯jℑVN. The other situation, there exists a Njv such that Njv∩VM∪VN∈ℑ. By the definition of ideal, Njv∩VM∈ℑ or Njv∩VN∈ℑ. Then, we have v∈N¯jℑVM or v∈N¯jℑVN. Hence, N¯jℑVM∪VN⊆N¯jℑVM∪N¯jℑVN.(e) We obtain thatN¯jℑVM∩VN⊆N¯jℑVM∩N¯jℑVN by (b). Let v∈N¯jℑVM∩N¯jℑVN. Then, v∈N¯jℑVM and N¯jℑVN. By the definition of ℑNj-lower approximations, we obtain Njv∩VM′∈ℑ and Njv∩VN′∈ℑ. By the definition of ideal, v∈N¯jℑVM∩VN. So N¯jℑVM∩VN⊆N¯jℑVM∩N¯jℑVN.
The following example shows that the equations in Proposition2 (d) and (e) may not be true in general. Also, it indicates that the converse implications of Proposition 2 (b) and (f) may not be provided.Example 2.
LetG be a graph as shown in Figure 1 and ℑ=∅,v2,v3,v2,v3.(a)
LetVH=v1,v3 and VM=v4,v5. N¯rℑVH∪VM=VG. N¯rℑVH=v3,v4,v5 and N¯rℑVM=v1,v5, so N¯jℑVH∪VM≠N¯jℑVH∪N¯jℑVM.(b)
LetVH=v3 and VM=v4,v5. Then, N¯rℑVH=v5. We obtain N¯jℑVH⊆N¯jℑVM but it is not VH⊆VM.(c)
LetVH=v1,v3 and VM=v4,v5. N¯rℑVH∩VM=∅. N¯rℑVH=v1,v2,v3,v4 and N¯rℑVM=v1,v2,v4,v5. So N¯jℑVH∩VM≠N¯jℑVH∩N¯jℑVM.(d)
LetVH=v1,v2,v5. Then, N¯rℑVH=VG but VH′∈ℑ.Remark 2.
The following properties which are given Proposition1 as Pawlak’s rough set model are not always hold.(a)
N¯jℑVM⊆VM(b)
N¯jℑN¯jℑVM=N¯jℑVM; N¯jℑN¯jℑVM=N¯jℑVM(c)
N¯jℑVM′=N¯jℑVM′; N¯jℑVM′=N¯jℑVM′Example 3.
LetG be a graph as shown in Figure 1 and ℑ=∅,v2,v3,v2,v3.(a)
LetVM=v1,v3. Then, N¯rℑVM=v3,v4,v5. So N¯rℑVM⊆VM.(b)
LetVM=v4,v5. Then, N¯rℑVM=v1,v2,v4,v5. N¯rℑN¯rℑVM=VG. So N¯rℑN¯rℑVM=VG≠N¯rℑVM.(c)
LetVM=v1,v2,v5. Then, N¯rℑVM=VG and N¯rℑVM′=v1,v5. So N¯jℑVM′≠N¯jℑVM′.Proposition 3.
The followings hold:(a)
N¯uℑVH⊆N¯rℑVH⊆N¯iℑVH(b)
N¯uℑVH⊆N¯lℑVH⊆N¯iℑVH(c)
N¯uℑVH⊆N¯rℑVH⊆N¯iℑVH(d)
N¯uℑVH⊆N¯lℑVH⊆N¯iℑVHProof.
It is obvious.Proposition 4.
The followings hold:(a)
N¯iℑVA⊆N¯rℑVA⊆N¯uℑVA(b)
N¯iℑVA⊆N¯lℑVA⊆N¯uℑVA(c)
N¯iℑVA⊆N¯rℑVA⊆N¯uℑVA(d)
N¯iℑVA⊆N¯lℑVA⊆N¯uℑVADefinition 10.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the ℑNj-boundary regions and ℑNj-accuracy of approximations of VH are defined, respectively, by(1)BNiℑVH=N¯jℑVH−N¯NjℑVH,σNjℑVH=N¯jℑVHN¯jℑVH,where|N¯jℑVH≠∅.Corollary 1.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, the followings hold:(a)
BNiℑVH⊆BNrℑVH⊆BNuℑVH(b)
BNiℑVH⊆BNlℑVH⊆BNuℑVH(c)
BNiℑVH⊆BNrℑVH⊆BDPNuℑVH(d)
BNiℑVH⊆BNlℑVH⊆BDPNuℑVHCorollary 2.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, the followings hold:(a)
σNuℑVH≤σNrℑVH≤σNiℑVH(b)
σNuℑVH≤σNlℑVH≤σNiℑVH(c)
σNuℑVH≤σNrℑVH⊆σNiℑVH(d)
σNuℑVH≤σNlℑVH⊆σNiℑVHLemma 1.
LetG be a graph, H be a subgraph of G, ℑ and J be ideals on G. Then, if ℑ⊆J, the followings hold:(a)
N¯jℑVH⊆N¯jJVH(b)
N¯jJVH⊆N¯jℑVH(c)
σNjℑVH≤σNjJVHCorollary 3.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the followings hold:(a)
BNiℑVH⊆BNiVH(b)
σNiVH≤σNiℑVHDefinition 11.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the ℑNj∗-lower and ℑNj∗-upper approximations of H are defined respectively by(a)
N∗¯jℑVH=VH∩N¯jℑVH(b)
N∗¯jℑVH=VH∪N¯jℑVHProposition 5.
LetG be a graph and H be subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the followings hold:(a)
N∗¯jℑVH⊆VH⊆N∗¯jℑVH(b)
N∗¯jℑ∅=N∗¯jℑ∅=∅Definition 12.
For everyj∈J, the ℑNj-boundary regions and ℑNj-accuracy of approximations of VH are described, respectively, by(2)BNi∗ℑVH=N∗¯jℑVH−N∗¯jℑVH,σNj∗ℑVH=N∗¯jℑVHN∗¯jℑVH,whereN∗¯jℑVH≠∅.Definition 13.
For everyj∈J, a subgraph H is called HNjℑ-definable (HNjℑ-exact) if N∗¯jℑVM=N∗¯jℑVM. Otherwise, H is called HNjℑ-rough set.Lemma 2.
LetG=VG,EG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, a subgraph H is HNjℑ-exact iff BNi∗ℑVH=∅.Example 4.
LetG be a graph as shown in Figure 1 and ℑ=∅,v2,v5,v2,v5. Let VH=v1,v2,v3,v4. Then, VH is HNjℑ-exact.Remark 3.
In Table1, the lower approximations, upper approximations, boundary regions, and accuracy measures are calculated by using El Atik et al.’s approach [28] and our approach according to the Example 2 taking the ideal ℑ=∅,v2,v5,v2,v5. Thus, it is seen that the accuracy measure of our approach is more precise than El-Atik et al.’s approach.Table 1
Comparison between the lower approximations, upper approximations,j-boundary, and j-accuracy measures by using El Atik et al.’s approach [28] and our approach for j=u.
V(H)El Atik et al.’s approach [28]Our approachN¯uVHN¯uVHBNuVHσNuVHN¯uℑVHN¯uℑVHBNuℑVHσNuℑVHv1∅v1,v2,v3,v4v1,v2,v3,v40∅v1,v2,v3,v4v1,v2,v3,v40v5∅v2,v5v2,v50v5v5∅1v2,v5v5VGv1,v2,v3,v41/5v5v2,v5v21/2v4,v5∅v1,v2,v4,v5v1,v2,v4,v50v5v1,v2,v4,v5v1,v2,v41/4v2,v3,v5v5VGv1,v2,v3,v41/5v5v1,v2,v4,v5v1,v31/4


## 4.I-P-J Approximation

In this section, the concepts ofℑPj-lower and ℑPj-upper approximations are given by using Pj-neighbourhoods for j∈J and ideal. Their essential properties and relationships among them are also examined. Besides, the accuracy measure of a subgraph is defined by using them. Also, it is proven that ℑPj-accuracy measure is the highest in the case of j=i, and it is shown that ℑPj-accuracy measures are the highest for j∈J when we compare them with ℑNj-accuracy measures and Pj-accuracy measures.Definition 14.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the ℑPj-lower and ℑPj-upper approximations of H are defined, respectively, by(a)
P¯jℑVH=v∈VG:Pjv∩VH′∈ℑ(b)
P¯jℑVH=VH∪v∈VG:Pjv∩VH∈ℑTheorem 2.
For everyj∈J, the followings hold:(a)
P¯jVH⊆P¯jℑVH(b)
P¯jℑVH⊆P¯jVHProof.
(a)
Letv∈P¯jVH. Then, Pjv⊆VH. Therefore, we obtain Pjv∩VH′=∅∈ℑ. Hence, v∈P¯jℑVH.(b)
The proof is similar to (a).Remark 4.
Ifℑ=∅ in Definition 14, P¯jℑVH(P¯jℑ(VH)) coincides with P¯jVHP¯jVH for each j∈J.Example 5.
LetG be a graph as shown in Figure 2 and ℑ=∅,v4,v5,v4,v5. VH=v2,v3,v4. Then,(a)
P¯uVH=v3 and P¯uℑVH=v2,v3,v5. Then, P¯uℑVH≠P¯uVH.(b)
P¯lVH=v2,v3,v4,v5P¯lℑVH=v2,v3,v4. Then, P¯lℑVH≠P¯lVH.Figure 2
A directed graphG.Proposition 6.
For everyj∈J, the followings hold:(a)
VM⊆P¯jℑVM(b)
VM⊆VN implies P¯jℑVM⊆P¯jℑVN; VM⊆VN implies P¯jℑVM⊆P¯jℑVN(c)
P¯jℑ∅=∅, P¯jℑVG=P¯jℑVG=VG(d)
P¯jℑVM∪VN⊇P¯jℑVM∪P¯jℑVN; P¯jℑVM∪VN=P¯jℑVM∪P¯jℑVN(e)
P¯jℑVM∩VN=P¯jℑVM∩P¯jℑVN; P¯jℑVM∩VN⊆P¯jℑVM∩P¯jℑVN(f)
IfVM′∈ℑ, then P¯jℑVM=VG; if VM∈ℑ, then P¯jℑVM=VMProof.
The proof is similar to Proposition2.
The following example shows that the equations in Proposition9 (d) and (e) may not be true in general. Also, it indicates that the converse implications of Proposition 9 (b) and (f) may not be provided.Example 6.
LetG be a graph as shown in Figure 1 and ℑ=∅,v2,v3,v2,v3.(a)
LetVH=v1,v3 and VM=v4,v5. P¯rℑVH∪VM=VG. P¯rℑVH=v1,v2 and P¯rℑVM=v2,v5, so P¯jℑVH∪VM≠P¯jℑVH∪P¯jℑVM.(b)
LetVH=v1,v2 and VM=v4,v5. Then, P¯rℑVH=v1,v2 and P¯rℑVM=v1,v2,v3,v4 So P¯jℑVH⊆P¯jℑVM, but it is not VH⊆VM.(c)
LetVH=v1,v3 and VM=v4. Then, P¯rℑVH∩VM=∅, P¯rℑVH=v1,v3,v4, and P¯rℑVM=v3,v4. So P¯jℑVH∩VM≠P¯jℑVH∩P¯jℑVM.(d)
LetVH=v2,v5. Then, P¯rℑVH=VH, but VH∈ℑ.Remark 5.
The following properties, which are given Proposition1 as Pawlak’s rough set model, are not always hold.(a)
P¯jℑVH⊆VH(b)
P¯jℑP¯jℑVH=P¯jℑVH; P¯jℑP¯jℑVH=P¯jℑVH(c)
(P¯jℑVM)′=P¯jℑVM′; (P¯jℑVM)′=P¯jℑVM′Example 7.
LetG be a graph as shown in Figure 1 and ℑ=∅,v2,v3,v2,v3.(a)
LetVH=v4,v5. Then, P¯rℑVH=v2,v5. So P¯rℑVH⊆VH.(b)
LetVH=v3,v5. Then, P¯rℑVM=v2,v5. P¯rℑP¯rℑVM=v2,v3,v4,v5. So P¯rℑP¯rℑVM≠P¯rℑVM.(c)
LetVM=v1,v2,v4. Then, P¯rℑVM=v1,v2,v3,v4 and P¯rℑVM′=v2,v5. So P¯jℑVM′≠P¯jℑVM′.Proposition 7.
The followings hold:(a)
P¯uℑVH⊆P¯rℑVH⊆P¯iℑVH(b)
P¯uℑVH⊆P¯lℑVH⊆P¯iℑVH(c)
P¯uℑVH⊆P¯rℑVH⊆P¯iℑVH(d)
P¯uℑVH⊆P¯lℑVH⊆P¯iℑVHProposition 8.
The followings hold:(a)
P¯iℑVH⊆P¯rℑVH⊆P¯uℑVH(b)
P¯iℑVH⊆P¯lℑVH⊆P¯uℑVH(c)
P¯iℑVH⊆P¯rℑVH⊆P¯uℑVH(d)
P¯iℑVH⊆P¯lℑVH⊆P¯uℑVHDefinition 15.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, the ℑNj-boundary regions and ℑPj-accuracy of approximations of VH are defined, respectively, by(3)BPiℑVH=P¯jℑVH−P¯jℑVH∩VH,σPjℑVH=P¯jℑVH∩VHP¯jℑVH,whereP¯jℑVH≠∅.
Ifℑ=∅, then BPiℑVH=BPiVH and σPjℑVH=σPjVH.Theorem 3.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈r,l,r,l,i,u,i,u, the followings hold:(a)
BPiℑVH⊆BPiVH(b)
σPjVH≤σPjℑVHCorollary 4.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, the followings hold:(a)
BPiℑVH⊆BPrℑVH⊆BPuℑVH(b)
BPiℑVH⊆BPlℑVH⊆BPuℑVH(c)
BPiℑVH⊆BPrℑVH⊆BPuℑVH(d)
BPiℑVH⊆BPlℑVH⊆BPuℑVHCorollary 5.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, the followings hold:(a)
σPuℑVH≤σPrℑVH≤σPiℑVH(b)
σPuℑVH≤σPlℑVH≤σNiℑVH(c)
σPuℑVH≤σPrℑVH≤σNiℑVH(d)
σPuℑVH≤σPlℑVH≤σNiℑVHLemma 3.
LetG be a graph, H be a subgraph of G, ℑ and J be ideals on G. Then, if ℑ⊆J the followings hold:(a)
P¯jℑVH⊆P¯jJVH(b)
P¯jJVH⊆P¯jℑVH(c)
σPjℑVH≤σPjJVHDefinition 16.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, a subgraph H is called HPjℑ-definable (HPjℑ-exact) if BPjℑVH=∅. Otherwise, H is called HPjℑ-rough set.Lemma 4.
LetG be a graph, H be a subgraph of G and ℑ be an ideal on G. Then, for every j∈J, every HPj-exact is HPjℑ-exact. (equivalently, every HPjℑ-rough is HPj-rough.)Proof.
LetVH be HPj-exact. Then, P¯jVH−P¯jVH=∅ and so BPjVH=∅. We obtain BPjℑVH=∅. Hence, VH is HPjℑ-exact.
The following example shows that the converse implication of Lemma4 may not be true in general.Table 2
Comparisons between thej-boundary and j-accuracy measures by using our approach for j=r,l,u,i.
V(H)Forj=rForj = lForj = uForj = iBPrℑVHσPrℑVHBPlℑVHσPlℑVHBPuℑVHσPuℑVHBPiℑVHσPiℑVHv1v1,v40∅1v1,v40∅1v2v20∅1v20∅1v3∅1∅1∅1∅1v4v10v51/2v1,v4,v50∅1v5v20v4,v50v2,v4,v50∅1v1,v2v1,v2,v40∅1v1,v2,v40∅1v1,v3v1,v41/3∅1v1,v41/3∅1v1,v4∅1v4,v51/3v4,v51/3∅1v1,v5v1,v2,v41/4v4,v51/3v1,v2,v4,v50∅1v2,v3v21/2∅1v21/2∅1v2,v4v1,v2,v4,v50v4,v51/3v1,v2,v4,v50∅1v2,v5∅1v4,v51/3v4,v51/3∅1v3,v4v1,v41/3v4,v51/3v1,v4,v51/4∅1v3,v5v22/3v4,v51/3v2,v4,v51/4∅1v4,v5v1,v2,v41/4∅1v1,v2,v41/4∅1v1,v2,v3v1,v2,v41/4∅1v1,v2,v41/4∅1v1,v2,v4v22/3v4,v51/2v2,v4,v51/4∅1v1,v2,v5v1,v41/2v4,v51/2v1,v4,v51/4∅1v1,v3,v4∅1v4,v51/2v4,v51/2∅1v1,v3,v5v2,v3,v42/5v2,v4,v52/5v1,v2,v4,v51/5∅1v1,v4,v5v23/4∅1v23/4∅1v2,v3,v4v1,v2,v41/4v4,v51/2v1,v2,v4,v51/5∅1v2,v3,v5∅1v4,v51/2v4,v51/2∅1v2,v4,v5v1,v41/2∅1v1,v41/2∅1v3,v4,v5v1,v2,v42/5∅1v1,v2,v42/5∅1v1,v2,v3,v4v23/4v4,v5(3/5)v2,v4,v52/5∅1v1,v2,v3,v5v1,v43/5v4,v53/5v1,v4,v52/5∅1v1,v3,v4,v5v24/5∅1v24/5∅1v2,v3,v4,v5v1,v43/5∅1v1,v43/5∅1Table 3
Comparisons between the lower approximations, upper approximations,j-boundary, and j-accuracy measures by using El Atik et al.’s approach [28] and our approach for j=r.
V(H)El Atik et al.’s approach [28]Our approachP¯rVHP¯rVHBPrVHσPrVHP∗¯rℑVHP∗¯rℑVHBPrℑVHσPrℑVHv1v1v1∅1v1v1∅1v2v2v2∅1v2v2∅1v3∅v3,v4v3,v40∅v3v30v4∅v3,v4v3,v40v4v3,v4v31/2v5v5v5∅1v5v5∅1v1,v2v1,v2v1,v2∅1v1,v2v1,v2∅1v1,v3v1v1,v3,v4v3,v41/3v1v1,v3,v4v3,v41/3v1,v4v1v1,v3,v4v3,v41/3v1,v4v1,v3,v4v42/3v1,v5v1,v5v1,v5∅1v1,v5v1,v5∅1v2,v3v2v2,v3,v4v3,v41/3v2v2,v3,v4v3,v41/3v2,v4v2v2,v3,v4v3,v41/3v2,v4v2,v3,v4v32/3v2,v5v2,v5v2,v5∅1v2,v5v2,v5∅1v3,v4v3,v4v3,v4∅1v3,v4v3,v4∅1v3,v5v5v3,v4,v5v3,v41/3v5v3,v5v51/2v4,v5v5v3,v4,v5v3,v41/3v4,v5v3,v4,v5v32/3v1,v2,v3v1,v2v1,v2,v3,v4v3,v41/2v1,v2v1,v2,v3v32/3v1,v2,v4v1,v2v1,v2,v3,v4v3,v41/2v1,v2,v4v1,v2,v3,v4v33/4v1,v2,v5v1,v2,v5v1,v2,v5∅1v1,v2,v5v1,v2,v5∅1v1,v3,v4v1,v3,v4v1,v3,v4∅1v1,v3,v4v1,v3,v4∅1v1,v4,v5v1,v5v1,v3,v4,v5v3,v41/2v1,v4,v5v1,v3,v4,v5v33/4v1,v3,v5v1,v5v1,v3,v4,v5v3,v41/2v1,v3v1,v3,v5v52/3v2,v3,v4v2,v3,v4v2,v3,v4∅1v2,v3,v4v2,v3,v4∅1v2,v3,v5v2,v5v2,v3,v4,v5v3,v41/2v2,v5v2,v3,v5v32/3v3,v4,v5v3,v4,v5v3,v4,v5∅1v3,v4,v5v3,v4,v5∅1v1,v2,v3,v4v1,v2,v3,v4v1,v2,v3,v4∅1v1,v2,v3,v4v1,v2,v3,v4∅1v1,v2,v3,v5v1,v2,v5VGv3,v43/5v1,v2,v5v1,v2,v3,v5v33/4v1,v3,v4,v5v1,v3,v4,v5VGv24/5v1,v3,v4,v5v1,v3,v4,v5∅1v2,v3,v4,v5v2,v3,v4,v5v2,v3,v4,v5∅1v2,v3,v4,v5v2,v3,v4,v5∅1Table 4
Comparisons between the lower approximations, upper approximations,j-boundary, and j-accuracy measures by using our approaches for j=l.
V(H)Our approach according toNjOur approach according toPjN∗¯lVHN∗¯lVHBNl∗ℑVHσNl∗ℑVHP∗¯lℑVHP∗¯rℑVHBPlℑVHσPlℑVHv1∅v1,v3,v4v1,v3,v40v1v1∅1v2∅v2v20v2v2∅1v4∅v1,v2,v4v1,v2,v40v4v3,v4∅1/2v5v5v2,v5v21/2v5v5∅1v2,v5v5v2,v5v21/2v2,v5v2,v5∅1v3,v4∅v1,v2,v3,v4v1,v2,v3,v40v3,v4v3,v4∅1v1,v2,v3v3v1,v2,v3,v4v1,v2,v31/4v1,v2v1,v2,v3,v4v3,v41/2v2,v3,v5v5VGv1,v2,v3,v41/5v2,v5v2,v3,v5v32/3v1,v2,v3,v4v1,v3,v4v1,v2,v3,v4v23/4v1,v2,v3,v4v1,v2,v3,v4∅1Example 8.
letG be a graph as shown in Figure 2 and ℑ=∅,v1,v4,v1,v4. Let VH=v1,v3. Then, VH is HPrℑ-exact, but it is not HPr-exact.Definition 17.
For everyj∈J, the ℑPj∗-lower and ℑPj∗-upper approximations of H are defined, respectively, by(a)
P∗¯jℑVH=VH∩P¯jℑVH(b)
P∗¯jℑVH=VH∪P¯jℑVHProposition 9.
For everyj∈J, the followings hold:(a)
P∗¯jℑVM⊆VM⊆P∗¯jℑVM(b)
P∗¯jℑ∅=P∗¯jℑ∅=∅Remark 6.
(a)
In Table2, the boundary regions and accuracy measures are calculated by using our approach as Example 5. Thus, it is seen that ℑPi-accuracy is more precise than ℑPj-accuracy for j=r,l,u.(b)
In Table3, the lower approximations, upper approximations, boundary regions, and accuracy measures are calculated by using El Atik et al.’s approach [28] and our approach according to the Example 5. Thus, it is seen that the accuracy measure of our approach is more precise than El Atik et al.’s approach.(c)
In Table4, the lower approximations, upper approximations, boundary regions, and accuracy measures are calculated by using our approach with according to the Example 2 and our approach according to the Example 5. Thus, it is seen that the accuracy measure of our approaches is incomparable.


## 5. Practical Examples

The main purpose of this section is to present a simple practice example to compare Atik et al.’s approaches and one of our approaches. We use Walczak’s example in Chemistry. LetVG=v1,v2,v3,v4,v5 be a five amino acids which represented in terms of five attributes: a1 = PIE, a2 = PIF, a3 = SAC = surface area, a4 = MR = molecular refractivity, a5 = Vol = molecular volume [30].Example 9.
Consider Table5 containing data about 5 persons.
We have five relations described asRi=vi,vj:viam−vjam≤σm/2,i,j=1,…,5,m=1,…,5 where σm represents the standard of the quantitative attributes am, m = 1…5. We show graph G in Figure 3 such that xR=∩xRk, k = 1,2,…,5 which uses to obtain j-neighborhood system for adjacent vertices. So, we obtain Nrv1=v1,v4, Nrv2=v2, Nrv3=v3, Nrv4=v1,v4, Nrv5=v5. Then, Prv1=Prv4=v1,v4, Prv2=v2, Prv3=v3, and Prv5=v5.
Then, taking the idealℐ=∅,v1,v4,v1,v4, we get Table 6.
From this practical example, accuracy measures of our approach are higher than El Atik’s approach [28]. Hence, these approaches are very useful in rough set theory.Table 5
Quantitative attributes of five amino acids.
a1a2a3a4a5v1−0.11−0.223353.458127.5v20.170.26282.92.7480105.3v300224.91.66265v40.150.13337.23.856140.6v51.281.73243.5180131.5Figure 3
G.Table 6
Comparisons between El Atik’s approach [28] approach and our approaches.
El Atik’s approach [28]Our approachBoundaryAccuracyBoundaryAccuracyv1v1,v40∅1v1,v2v1,v41/3∅1v2,v5∅1∅1v4,v5v1,v41/3∅1v1,v2,v3v1,v41/2∅1v3,v4,v5v1,v41/2∅1v1,v2,v3,v5v1,v43/5∅1


## 6. Conclusion

Graph theory and rough set theory are being studied by a lot of researchers, and they have many applications. Neighbourhood systems based on vertices of graphs are studied by a lot of researchers. Then, the neighbourhood systems and rough sets on graphs are used in many different fields such as physics and medicine.In this paper, we generated different approximations of a graph on neighbourhood systems by using ideals. A class of approximations calledℑNj-approximations (ℑPj-approximations) where j∈J by using ideal. Then, we investigated the relationships between them and formulated the concept of ℑNj-boundary region (ℑPj-boundary region) and ℑNj-accuracy of approximations (ℑPj-accuracy of approximations) of a subgraph. Moreover, we compared our approximations with the ones defined before and see that these approximations are more accurate. Finally, we used Walczak’s example in chemistry and showed that our approach is better than the one before on a real-life chemistry problem.In future work, one can be focused on new types of approximations to obtain higher accuracy measures than those given in the literature by using different neighbourhoods such asSj-neighbourhoods [12] and maximal rough neighbourhoods [13] on graphs. Also, the methods that we obtained can be applied to various real-life problems.



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