#### Abstract

In this paper, we generate different approximations for by using the concept of ideal, -neighbourhoods and -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.

#### 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 -approximations in the -neighbourhood space. Atef et al. [10] gave the generalization of some types of rough set models based upon -neighbourhood space and other types of rough set models based on -adhesion neighbourhood space. They also introduced the notions of -neighbourhoods using j-neighbourhoods and investigated their properties. Then, Al-Shami [11] established the notions of -neighbourhoods using j-neighbourhoods and studied their characteristics. Also, Al-Shami and Ciucci [12] studied the notions of -neighbourhoods using j-neighbourhoods and investigated their properties. Lastly, Al-Shami [13] defined and studied a new class of neighbourhoods system which is called -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 -neighbourhoods. Later, Hosny [16] generated different topologies using the concept of the ideal. Hosny constructed new approximations namely --approximations using these topologies. Also, she proved that these approximations are extended the notation of -approximations. Hosny [17] presented the concepts of -nearly open sets with respect to ideals and -nearly approximations in terms of ideals as generalizations of -nearly approximations. Then, Hosny et al. [18] produced some topologies by using the concept of ideal and -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, -neighbourhoods and -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 is a pair of vertices sets and edges sets . 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 be a graph; we call a subgraph of *G* if and , and under the circumstances, we write . 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 Section 2, 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 ---approximations, where 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 ---boundary region and ---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 ---lower and ---upper approximations, ---boundary region, and ---accuracy measure of a subgraph of by using ideals and -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]). *Let be a nonempty set. Then, a family of sets is said to be an ideal in if(a) imply (b) and imply

*Definition 2 (see [7]). *Let be a binary relation on a nonempty set *U* and . Lower approximations and under approximations of *X* are defined by(a)(b), where

Proposition 1 (see [1, 2]). *Let be a binary relation on a nonempty set U and . Then, the following properties hold:*

*Definition 3 (see [7, 28, 29]). *Let be a graph for each . The -neighbourhood of , is defined as(a)(b)(c)(d)(e)(f)(g)(h)

*Definition 4 (see [28]). *Let be a graph and be a subgraph of and . The first-type lower *j*-approximations and upper *j-*approximations are defined, respectively, by(a)(b)

*Definition 5 (see [28]). *Let be a graph and be a subgraph of and . -boundary regions and -accuracy measures of in terms of *j*-adhesion neighbourhood are defined, respectively, by(a)(b), where

*Definition 6 (see [10]). *Let be a graph for each . Then, the -neighbourhoods of , are defined as(a)(b)(c)(d)(e)(f)(g)(h)

*Definition 7 (see [28]). *Let be a graph and be a subgraph of and . The first type lower *j*-approximations and upper *j*-approximations are defined, respectively, by(a)(b)

*Definition 8 (see [28]). *Let be a graph and be a subgraph of and . -boundary regions and -accuracy measure of in terms of *j*-adhesion neighbourhood are defined, respectively, by(a)(b), where

#### 3. *I*-*N*-*J* Approximation

In this section, the concepts of -lower and -upper approximations are given by using -neighbourhoods for 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 -accuracy measures are more precise when we compare them with -accuracy measures for .

*Definition 9. *Let be a graph, be a subgraph of and be an ideal on . Then, the -lower and -upper approximations of are defined, respectively, by(a)(b)

Theorem 1. *Let be a graph, be a subgraph of and be an ideal on . Then, for every , the followings hold:*(a)*(b)*

*Proof. *(a)Let . Then, . Therefore, we obtain . Hence, (b)The proof is similar to (a).

*Remark 1. *If in Definition 9, () coincides with for each .

*Example 1. *Let *G* be a graph as shown in Figure 1 and .(a)Let . Then, and . So .(b)Let . Then, . So .

Proposition 2. *Let be a graph, and be subgraphs of and be an ideal on . Then, for every , the followings hold:*(a)*(b)** implies ; implies *(c)*, *(d)*; *(e)*; *(f)*If , then ; if , then *

*Proof. * (b) Let . Then, . Since and by the definition of ideal, . So . (d) We obtain that by (b). Let . Then, or . If , then we have . The other situation, there exists a such that . By the definition of ideal, or . Then, we have or . Hence, . (e) We obtain that by (b). Let . Then, and . By the definition of -lower approximations, we obtain and . By the definition of ideal, . So .The following example shows that the equations in Proposition 2 (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. *Let *G* be a graph as shown in Figure 1 and .(a)Let and . . and , so .(b)Let and . Then, . We obtain but it is not .(c)Let and . . and . So .(d)Let . Then, but .

*Remark 2. *The following properties which are given Proposition 1 as Pawlak’s rough set model are not always hold.(a)(b); (c);

*Example 3. *Let *G* be a graph as shown in Figure 1 and .(a)Let . Then, . So .(b)Let . Then, . . So .(c)Let . Then, and . So .

Proposition 3. *The followings hold:*(a)*(b)**(c)**(d)*

*Proof. *It is obvious.

Proposition 4. *The followings hold:*(a)*(b)**(c)**(d)*

*Definition 10. *Let be a graph, be a subgraph of and be an ideal on . Then, for every , the -boundary regions and -accuracy of approximations of are defined, respectively, by

Corollary 1. *Let be a graph, be a subgraph of and be an ideal on . Then, the followings hold:*(a)*(b)**(c)**(d)*

Corollary 2. *Let be a graph, be a subgraph of and be an ideal on . Then, the followings hold:*(a)*(b)**(c)**(d)*

Lemma 1. *Let be a graph, be a subgraph of , and be ideals on . Then, if , the followings hold:*(a)*(b)**(c)*

Corollary 3. *Let be a graph, be a subgraph of and be an ideal on . Then, for every , the followings hold:*(a)*(b)*

*Definition 11. *Let be a graph, be a subgraph of and be an ideal on . Then, for every , the -lower and -upper approximations of are defined respectively by(a)(b)

Proposition 5. *Let be a graph and be subgraph of and be an ideal on . Then, for every , the followings hold:*(a)*(b)*

*Definition 12. *For every , the -boundary regions and -accuracy of approximations of are described, respectively, by

*Definition 13. *For every , a subgraph is called -definable (-exact) if . Otherwise, is called -rough set.

Lemma 2. *Let be a graph, be a subgraph of and be an ideal on . Then, for every , a subgraph is -exact iff .*

*Example 4. *Let *G* be a graph as shown in Figure 1 and . Let . Then, is -exact.

*Remark 3. *In Table 1, 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 . Thus, it is seen that the accuracy measure of our approach is more precise than El-Atik et al.’s approach.

#### 4. *I*-*P*-*J* Approximation

In this section, the concepts of -lower and -upper approximations are given by using -neighbourhoods for 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 -accuracy measure is the highest in the case of , and it is shown that -accuracy measures are the highest for when we compare them with -accuracy measures and -accuracy measures.

*Definition 14. *Let be a graph, be a subgraph of and be an ideal on . Then, for every , the -lower and -upper approximations of are defined, respectively, by(a)(b)

Theorem 2. *For every , the followings hold:*(a)*(b)*

*Proof. *(a)Let . Then, . Therefore, we obtain . Hence, .(b)The proof is similar to (a).

*Remark 4. *If in Definition 14, coincides with for each .

*Example 5. *Let *G* be a graph as shown in Figure 2 and . . Then,(a) and . Then, .(b). Then, .

Proposition 6. *For every , the followings hold:*(a)*(b)** implies ; implies *(c)*, *(d)*; *(e)*; *(f)*If , then ; if , then *

*Proof. *The proof is similar to Proposition 2.

The following example shows that the equations in Proposition 9 (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. *Let *G* be a graph as shown in Figure 1 and .(a)Let and . . and , so