Abstract
Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.
1. Introduction
There are several techniques and methods to model real-life events. But, in practicality, the information is sometimes uncertain, unclear, varying, and indeterminate, which is difficult to model using usual methods. Different theories are proposed for modeling uncertainty. Smarandache [1] gave a novel concept of expressing and solving uncertainty. He initiated the neutrosophic sets (NSs), which are the extensions of fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). FSs proposed by Zadeh [2] and their generalizations are used to model the uncertainty. FSs involve the membership grade for each object whose value lies in the unit interval [0, 1]. The concept of IFSs was proposed and introduced by Atanassov in 1983 [3–5]. These are the generalization of FSs. The extension in IFSs is that in these sets the members of IFSs are assigned the membership grade as well as the nonmembership grade, provided that the sum of both grades lies in the unit interval [0, 1]. Atanassov also discussed the types, properties, and applications of IFSs. But these concepts are unable to model many real-life problems, so Smarandache [6] devised the notion of NSs that can deal with such problems. In NS, the object carries three independent grades: membership, abstinence, and nonmembership. Each of these grades lies in nonstandard unit interval ]−0, 1+[ and their sum lies in nonstandard interval ]−0, 3+[. Later the concept of single-valued neutrosophic set (SVNS) was introduced by Wang et al. [7], which is the subclass of NSs. This is even more practical and applies to a wide range of problems.
Graph theory is one of the major branches of mathematics and combinatorics. It has many applications in numerous fields such as computer science, networking, geometry, algebra, set theory, economics, medicine, engineering, and chemistry. Hence, it has a wide range of applications in real world. Cantor presented the notion of crisp graphs, which uses the concepts of classical set theory. These graphs cannot model the problems associated with uncertainty because, in crisp graph, there are only two possibilities for a vertex and an edge, that is, whether it belongs to the graph or does not. In other words, it only conveys that the objects are related or not but does not indicate the strength and weakness of the relationship. So, an extension of crisp graphs is a fuzzy graph (FG), which was introduced by Rosenfeld in 1975 [8]. The idea of fuzziness in graph theory was given by Kauffman [9] in 1973 by using the fuzzy relations. Rosenfeld [8] further introduced the concepts related to FG such as connectedness, paths, cycles, trees, and bridges. Moreover, he also described some of the properties of FGs. Later on, the work of Rosenfeld was extended by many researchers who discussed its types, properties, and applications in real life, such as regular fuzzy graphs [10], fuzzy tolerance graphs [11], bipolar fuzzy graphs [12–14], interval-valued fuzzy graphs, balanced interval-valued fuzzy graphs [15, 16], fuzzy planar graphs [17, 18], bipolar fuzzy hypergraphs [19, 20], and completeness and regularity of generalized fuzzy graphs [21].
One of the generalizations of FGs is intuitionistic fuzzy graph (IFG). The notion of IFG was presented by Shannon and Atanassov who also developed the idea of IFS relation, which was used in defining the IFGs [22]. They also discussed the many properties, theorems, and proofs regarding IFSs and IFGs. The work of Atanassov and Shannon was extended by many researchers who presented its properties, types, and applications; for example, join, union, and product of two IFGs were defined by Parvathi et al., and strong products, direct products, and lexicographic products of two IFGs were presented by Rashmanlou et al. The concepts of strong IFGs and intuitionistic fuzzy hypergraphs along with their applications were discussed by Akram and Davvaz [23, 24]. Karunambigai et al. discussed the balanced intuitionistic fuzzy graphs [25], covering and paired domination in IFGs [26], and intuitionistic fuzzy tolerance graph along with its applications was discussed by Sahoo and Pal [27, 28]. Even though the FGS and IFGs are great tools to model real-world problems, some of the problems with uncertain, indeterminate, and varying information cannot be handled by FGs and IFGs. Therefore, some more efficient tool was required to tackle such problems, and this requirement led to the idea of NSs and NGs.
FSs and FGs discuss only the membership grades of the objects, while IFSs and IFGs discuss both the membership and nonmembership grades of an object, which do not totally model the human opinions, as human opinion can also be abstentious. Moreover, in IFSs and IFGs, the membership and nonmembership grades cannot be chosen independently. Their sum must be restricted to be in the unit interval. Therefore, this void created the space for the introduction of NSs and NGs. Smarandache, after initiating the concept of NSs, proposed four main categories of NGs: I-vertex NG and I-edge NG are based on literal indeterminacy (I), while (t, I, f)-vertex NG and (t, I, f)-edge NG are based on (t, I, f) components [29–33]. In NSs and NGs, the membership, abstinence, and nonmembership grades can be chosen independently from the unit interval, which provides comfort and ease to a decision-maker. Lately, NGs caught the attention of many researchers [34–40]. In addition, Garg [41–43] introduced aggregation operators and multiple decision-making techniques via SVNSs.
This study focuses on the concepts of SVNGs, which were initiated by Broumi and Smarandache by applying the concept of SVNS to graphs [44]. They also discussed different types of SVNGs, bipolar SVNGs [45], grade, order, and size [46] and investigated many properties with proofs and examples. Recently, some researchers have worked on SVNGs, for instance, operations on SVNGs presented by Akram and Shahzadi [47], applications of operations on SVNGs given by Naz et al. [36], and properties of SVNGs discussed by Karaaslan and Davvaz [48].
Since the domain of SVNG is greater than that of FGs and IFGs, it expands the range of applications of graphs. Like all other graphs, SVNGs also model the relations. So, it is applied to all sorts of problems consisting of relationships. It is capable of modeling the problems with uncertain and varying information in real world, where FGs and IFGs fail. This article introduces the notions of covering, matching, and paired domination in SVNG, complete SVNGs, and complete bipartite SVNGs. Moreover, strong coverings, that is, strong vertex cover and strong edge cover, the strong vertex covering number, and strong edge covering numbers are defined using the strong edge. In addition, the strong independent sets, the strong independent number, strong matching, and strong matching number are also discussed along with some interesting properties, theorems, proofs, and examples. The paired domination in SVNGs is also explained with examples and theorems.
The paper is organized as follows: Section 2 reviews some basic definitions and examples of FGs, IFSs, IFRs, IFGs, SVNSs, SVNRs, and SVNGs. Section 3 defines the strong covering of vertices and edges, strong independent sets (SIS), and strong matchings (SM) using strong edges (SEs) along with suitable examples and several interesting properties. Section 4 introduces the paired domination (PD), strong paired domination (SPD), and perfect paired domination. Finally, the research is concluded in Section 5.
2. Preliminaries
This section recalls some basic definitions of fuzzy graph (FG), intuitionistic fuzzy set (IFS), intuitionistic fuzzy relation (IFR), intuitionistic fuzzy graph (IFG), single-valued neutrosophic set (SVNS), single-valued neutrosophic relation (SVNR), and single-valued neutrosophic graph (SVNG). In addition, the examples of graphs are also given for better understanding.
Definition 1. (see [8]). A pair is said to be an FG, where is an FS in universe Ẍ defined as and called the collection of vertices possessing the grade of membership and the collection on Ẍ is defined as
and called the collection of edges possessing the grade of membership , such that
Example 1. An illustration of an FG is given in Figure 1. Here, the collections of vertices and edges are and , respectively.
Definition 2. (see [3]). A nonempty collection A in universe Ẍ of the following form is called an IFS:where the mapping represents the grade of membership and the mapping represents the nonmembership of A, such that
Definition 3. (see [3]). Let and be two IFSs such that and are mappings representing the membership and nonmembership grades of , respectively, and and are mappings representing the membership and nonmembership grades of , respectively; then the intuitionistic fuzzy relation (IFR) denoted by Ṝ is a collection of the formThe mapping represents the membership grade and the mapping represents the nonmembership grade of , whereThis is provided that .
Definition 4. (see [22]). An IFG is a pair , where is an IFS in universe Ẍ defined as
and called the collection of vertices possessing the membership grade and nonmembership grade , such thatand the collection on Ẍ is defined as and called the collection of edges possessing the membership grade and the nonmembership grade , such thatThis is provided that .
Example 2. In Figure 2, the collection of vertices and the collection of edges for an IFG are and , respectively.
Definition 5. (see [7]). A nonempty collection A in universe Ẍ of the following form is called an SVNS:where the mapping represents the membership grade, represents the abstinence grade, and represents the nonmembership grade such that
Definition 6. (see [46]). Let and be two SVNSs; then, is said to be the SVNR on , if ,This is provided that , where , and are the membership, abstinence, and nonmembership grades of , respectively, and , and are the membership, abstinence, and nonmembership grades of , respectively.
Definition 7. (see [44]). An SVNG is a pair , where is an SVNS in Ẍ defined as and called the collection of vertices possessing the membership grade , abstinence grade , and nonmembership grade , such thatand is called the collection of edges possessing the membership grade , abstinence grade , and nonmembership grade , such thatThis is provided that .
Example 3. An SVNG is constructed in Figure 3 using the collection of vertices and the collection of edges .
3. Covering and Matching in Single-Valued Neutrosophic Graphs
This section presents the definitions and examples of strong covering of vertices and edges, strong independent sets (SIS), and strong matchings (SM) using strong edges (SEs). Some interesting theorems are also proved for covering and matching in complete SVNG, SVNC, and complete bipartite SVNG.
Definition 8. Let be an SVNG. A vertex and an SE incident to it are said to strongly cover each other. The collection of vertices Ṩ which covers all the SEs of an SVNG is known as strong vertex cover (SVC) in is the membership grade, is the abstinence grade, and is the nonmembership grade values of the SVC Ṩ, where and are the minimum of the membership and the abstinence values, respectively, and is the maximum value among the collection of the nonmembership values of the strong edges incident on . Let and denote the minimum values of membership and abstinence, and let denote the maximum value of nonmembership of the SVC of Then the strong vertex covering number of is defined and denoted by . An SVC with minimum membership value, minimum abstinence value, and maximum nonmembership value in an SVNG is said to be a minimum SVC.
Theorem 1. Let be a complete SVNG; then, , and are defined aswhere is the membership value, the abstinence value, and is the nonmembership value of the weakest edge in , and n is the number of vertices in.
Proof. Since is a complete SVNG, each of its vertices is connected to every other vertex in and all of its edges are strong. Hence, a strong cover vertex of is formed by any collection of vertices. Let be a minimum SVC, and is connected to the vertices . Then, the edges are all the weakest edges of with as the membership strength, as the abstinence strength, and as the nonmembership strength, where the collection of vertices forms an SVC of withwhere is the minimum of the membership value of the SEs incident on . If the value of membership for the weakest edge of graph is represented by , thenTherefore, .
Now,where is the minimum of the abstinence value of the SE incident on . If the value of abstinence for the weakest edge of graph is represented by , thenTherefore, .
In the same way,where is the maximum of the nonmembership value of the SE incident on . If the value of abstinence for the weakest edge of graph is represented by , thenTherefore, .
Theorem 2. If is a complete bipartite SVNG with and as partite sets, then
Proof. Since is a complete bipartite SVNG, all of its edges are strong. Moreover, each of the vertices in is connected to all the vertices in and each of the vertices in is connected to all the vertices of . The collection of all edges of is the union of the collection of all edges incident to every vertex in and the collection of all edges incident to each vertex in . Therefore, , , and are the SVCs in . It is obvious thatTherefore, .
And, and .
Therefore, .
Similarly, and .
Therefore, .
Theorem 3. If is a single-valued neutrosophic cycle (SVNC) and is a crisp cycle (CC), then
Proof. As every edge in an SVNC is strong, the SVC number of is , since the number of strong vertices in SVNG and the crisp cycle are the same because each edge is strong in both graphs [49]. Therefore, in the SVC of is the least number of vertices. So,
Definition 9. Two vertices are said to be strongly independent in an SVNG if they are not connected by an SE. If any collection in contains any two strongly independent vertices, then such a collection is known as strong independent set (SIS).
Definition 10. Let be an SIS in an SVNG . Then the values of membership, abstinence, and nonmembership of are defined as , , and , respectively, where and represent the least value among the values of membership and the least value among the values of abstinence of the SEs incident on . Similarly, the nonmembership value of is defined, where represents the greatest value among the values of nonmembership of the SEs incident on .
denotes and defines the strong independent number of an SVNG , where are greatest values of membership and greatest values of abstinence of in , respectively, and is the least value of nonmembership of in . The SIS with maximum membership value, maximum abstinence value, and minimum nonmembership value in an SVNG is known as a maximum SIS of vertices.
Theorem 4. If is a complete SVNG, then , , and , where , , and are the membership, abstinence, and nonmembership values of the weakest edge in .
Proof. Since is a complete SVNG, each of its vertices is connected to every other vertex in and all of its edges are strong. Therefore, there is only single SIS, that is, . Hence, the result follows.
Theorem 5. If is a complete bipartite SVNG with and as partite sets, then
Proof. Since is a complete bipartite SVNG, all of its edges are strong. Moreover, each of the vertices in is connected to all the vertices in , and each of the vertices in is connected to all the vertices of . Therefore, and are the SISs in . Hence,
Theorem 6. Let be an SVNC and denotes a CC, then
Proof. As every edge in an SVNC is strong, the SVC number of is , since the number of vertices in a SISs in and are the same because each edge is strong in both graphs [49]. So, in the SVC of is the greatest number of vertices. Thus,
Definition 11. Let be a connected SVNG. The collection Ṫ of SEs which covers all the vertices of an SVNG is known as strong edge cover in G. , , and are the membership, abstinence, and nonmembership values of SEC Ṫ, respectively. is the SEC number of an SVNG , where and are the minimum membership and abstinence values of the SEC of SVNG , whereas is the greatest value of nonmembership. An SEC with minimum membership value, minimum abstinence value, and maximum nonmembership value in an SVNG is said to be a minimum SEC.
Theorem 7. If is a complete SVNG, then
Proof. As each of the vertices in a complete SVNG is connected to every other vertex of and all of its edges are strong, the SEC number of is , since each edge is strong in a complete SVNG and crisp graph; therefore, the numbers of strong edges in both graphs are the same [49]. Hence, the least number of edges in an SEC in is . Hence,
Theorem 8. If is a complete bipartite SVNG with and as partite sets, then
Proof. Since is complete bipartite SVNG, all of its edges are strong. Moreover, each of the vertices in is connected to all the vertices in and vice versa. The edge covering number of is , since each edge is strong in a complete bipartite SVNG and ; therefore, the number of SEs in both graphs is the same [49]. Hence, the least number of edges in the SEC in is . Thus,
Theorem 9. Let be an SVNC and is a CC, then
Proof. Since every edge in an SVNC is strong, the SEC number of is , as the number of SEs in SVNG and the crisp cycle are the same because each edge is strong in both graphs [49]. So, in the SEC of , is the least number of edges. Thus,
Definition 12. A collection of SEs denoted by Ṏ in an SVNG is said to be an SIS of edges if all of its edges do not share a vertex. Ṏ is also known as strong matching (SM) in .
Definition 13. If , where is an SM in an SVNG , then it is said that is strongly matched to by . , , and are the membership, abstinence, and nonmembership values of the SEC , respectively.
is the strong edge independent number or SM number of an SVNG , where and are the maximum membership and abstinence values of the SMs of , and denotes the minimum nonmembership value. An SM with maximum membership value, maximum abstinence value, and minimum nonmembership value in an SVNG is said to be a maximum SM.
Theorem 10. Let be an SVNG; thenwhere n denotes the number of vertices in .
Proof. As each vertex of a complete SVNG is connected to every other vertex in and all of its edges are strong, the SM number of is , since each edge is strong in a complete SVNG and crisp graph; therefore the SM numbers in both graphs are the same [49]. So, in the SM of , is the greatest number of edges. Thus,
Theorem 11. If is a complete bipartite SVNG with and as partite sets, then
Proof. Since is a complete bipartite SVNG, all of its edges are strong. Moreover, each of the vertices in is connected to all the vertices in and each of the vertices in is connected to all the vertices in . The matching number of is , since each edge is strong in a complete bipartite SVNG and complete bipartite crisp graph; therefore, the SM numbers in both graphs are the same [49]. So, in the SM of , is the greatest number of edges. Thus,
Theorem 12. If is SVNC and is a CC, then
Proof. Every edge of an SVNC is strong. Moreover, is the SM number of , and since each edge is strong in both of and , both graphs possess the same number of edges in SM [49]. So, in the SM of , is the greatest number of edges. Thus,
Example 4. Figure 4 displays an SVNG . In , the edges represented by solid lines , and are the SEs, while is not an SE, which is represented by dotted line. So, the SVCs are , , , , , , , and .
Now, by the definition of the value of SVC,the following results are obtained for each SVC: ; similarly,The SVC number of is . In this case, there is no minimum strong independent cover. In this case, there is no minimum SVC.
The vertices and are strongly independent since they are not linked by an SE. But they are not independent because are they adjacent.
The SISs are , , and .
Now, the value of strong independent cover is calculated by usingThus, ; similarly,The strong independent number of is . The maximum SIS is with .
The SECs are and .
The value of SEC is calculated byConsequently, and are the values of SECs and , respectively. So, the SEC number is . None of the strong covers is a minimum SEC.
The only SM of is . The SM number is defined byThus, is the SM number.
Theorem 13. Let be an SVNG of order which does not contain any isolated vertices; then, for every such graph,(i)(ii)
Proof. Let be the minimum SVC of and . Then is an SIS of vertices. That means contains such vertices that are incident on SEs of . So, .Let be the maximum SIS of vertices in and . It means does not contain any vertices that are connected to each other via SE and therefore all the SEs of are strongly covered by the vertex in . Hence, is an SVC of with smallest value . Therefore, .Using (46) and (47),By definition, is as follows:Hence,As are determined by considering the value of the SEs and the sum of the vertex values is represented by , the second inequality follows directly.
Definition 14. Suppose that is an SM in an SVNG and if strongly matches all vertices of to some vertices of , then is said to be perfect strong matching (PSM).
Example 5. All of the edges of an SVNG are strong and are represented by solid lines, as shown in Figure 5. Therefore, the SMs are the collection of , , and . Moreover, each of the collections , and is also a PSM. The SM number is calculated byHence, the SM numbers areSimilarly, and .
4. Paired Domination in Single-Valued Neutrosophic Graph
This section introduces the concept of paired domination (PD), strong paired domination (SPD), and perfect paired domination. A couple of theorems are proved, and examples are also given to explain the concepts.
Definition 15. Suppose that denotes an SVNG and is a vertex in ; then, each of the neighbors of and itself is strongly dominated by , which means that the vertices in are strongly dominated by . If every vertex of is a strong neighbor of some vertex in of vertices of , then is a strong dominating collection of .
Definition 16. is the membership value, is the abstinence value, and is the nonmembership value of the strong dominating collection , where and represent the minimum of membership values and minimum of abstinence values of the SEs incident on , while represents the maximum of the values of nonmembership of such edges.
denotes and defines the strong domination number of an SVNG , where and represent the minimum membership value and minimum abstinence value of strong dominating collections of , while represents the maximum nonmembership value of such collections.
Definition 17. If a collection of vertices of an SVNG such that is a strong dominating collection and the induced single-valued neutrosophic subgraph (SVNSG) has a perfect strong matching, then is called a strong paired dominating collection. and are the membership value and abstinence value of the strong paired dominating collection , while is the nonmembership value of , where and represent the minimum values of the membership and minimum values of the abstinence of SEs incident on , while represents the maximum values of the nonmembership of such edges.
denotes and defines the SPD number of an SVNG , where and represent the minimum membership value and minimum abstinence value of strong paired dominating collections of , while represents the greatest nonmembership value of such sets.
Example 6. Figure 6 displays an SVNG . In , the edges represented by solid lines , and are the SEs, while is not a SE, which is represented by a dotted line. So, the strong dominating collections are , , and .
Now, the values of strong dominating collections are to be found byTherefore, .
Similarly, and .
The SPD number is defined asThus, is the SPD number of SVNG in Figure 6.
Theorem 14. If is a complete SVNG, then , , and , where , , and are the membership, abstinence, and nonmembership values of the weakest edge in .
Proof. Since, by definition of a complete SVNG , each vertex of is connected to every other vertex in and all of its edges are strong, let be two vertices in any collection of , then a strong paired dominating collection is formed by such set. Thus, , , and .
Theorem 15. The SPD numbers for a complete bipartite SVNG are , , and , where , , and are the membership, abstinence, and nonmembership values of the weakest edge in .
Proof. By definition of , all of its edges are strong. Moreover, each of the vertices in is connected to all the vertices in . Therefore, any collection in which contains two vertices, one in and the other in , is a strong paired dominating collection. Let be the end vertices of any weakest edge in such that and ; then makes a strong paired dominating collection.
Thus, , , and .
5. Conclusion
Single-valued neutrosophic sets are characterized by three independent fuzzy-valued mappings, that is, membership grade, abstinence grade, and nonmembership grade. This research presented the notions of covering, matching, and paired domination using the strong edges in the neutrosophic environment. Furthermore, this paper discussed the concepts of strong vertices, strong edges, strong vertex cover, strong edge cover, strong vertex covering number, strong edge covering number, strong independent number, strong matching, and strong paired domination number in SVNG by strong edges with suitable examples. Moreover, the strong independent number, the strong matching, and the strong paired domination number of complete SVNG and complete bipartite are determined. The relations among these notions and numerous interesting properties are also proposed. Many interesting results of covering, matching, and paired domination are obtained and proved for SVNGs, complete SVNGs, SVNC, and complete bipartite SVNGs.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported by the Major Project of Education Department in Hebei (No. ZD2021039) and Innovation and Improvement Project of Academic Team of Hebei University of Architecture (Mathematics and Applied Mathematics) (No. TD202006).