Abstract
Many problems of practical interest can be modeled and solved by using vague graph (VG) algorithms. Vague graphs, belonging to the fuzzy graphs (FGs) family, have good capabilities when faced with problems that cannot be expressed by FGs. Hence, in this paper, we introduce the notion of - HMs of VGs and classify homomorphisms (HMs), weak isomorphisms (WIs), and coweak isomorphisms (CWIs) of VGs by - HMs. Hospitals are very important organizations whose existence is directly related to the general health of the community. Hence, since the management in each ward of the hospital is very important, we have tried to determine the most effective person in a hospital based on the performance of its staff.
1. Introduction
Graphs, from ancient times to the present day, have played a very important role in various fields, including computer science and social networks, so that with the help of the vertices and edges of a graph, the relationships between objects and elements in a social group can be easily introduced. But there are some phenomena in our lives that have a wide range of complexities that make it impossible for us to express certainty. These complexities and ambiguities were reduced with the introduction of FSs by Zadeh [1]. Since then, the theory of FSs has become a vigorous area of research in different disciplines including logic, topology, algebra, analysis, information theory, artificial intelligence, operations research, and neural networks and planning [2–6]. The FS focuses on the membership degree of an object in a particular set. But membership alone could not solve the complexities in different cases, so the need for a degree of membership was felt. To solve this problem, Gau and Buehrer [7] introduced false-membership degrees and defined a VS as the sum of degrees not greater than 1. The first definition of FGs was proposed by Kafmann [8] in 1993, from Zade’s fuzzy relations [9, 10]. But Rosenfeld [11] introduced another elaborated definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts such as paths, cycles, and connectedness. Ramakrishna [12] introduced the concept of VGs and studied some of their properties. Akram et al. [13–16] defined the vague hypergraphs, Cayley-VGs, and regularity in vague intersection graphs and vague line graphs. Rashmanlou et al. [17] investigated categorical properties in intuitionistic fuzzy graphs. Bhattacharya [18] gave some remarks on FGs, and some operations of FGs were introduced by Mordeson and Peng [19]. The concepts of weak isomorphism, coweak isomorphism, and isomorphism between FGs were introduced by Bhutani in [2]. Khan et al. [20] studied vague relations. Talebi [21, 22] investigated Cayley-FGs and some results in bipolar fuzzy graphs. Borzooei [23] introduced domination in VGs. Ghorai and Pal studied some isomorphic properties of m-polar FGs [24]. Jiang et al. [25] defined vertex covering in cubic graphs. Krishna et al. [26] presented a new concept in cubic graphs. Rao et al. [27–29] investigated dominating set, equitable dominating set, and isolated vertex in VGs. Hoseini et al. [30] given maximal product of graphs under vague environment. Jan et al. [31] introduced some root-level modifications in interval-valued fuzzy graphs. Amanathulla et al. [32] defined new concepts of paths and interval graphs. Muhiuddin et al. [33, 34] presented the reinforcement number of a graph and new results in cubic graphs.
A VG is a generalized structure of an FG that provides more exactness, adaptability, and compatibility to a system when matched with systems run on FGs. Also, a VG is able to concentrate on determining the uncertainty coupled with the inconsistent and indeterminate information of any real-world problems, where FGs may not lead to adequate results. VGs have a wide range of applications in the field of psychological sciences as well as in the identification of individuals based on oncological behaviors. Thus, in this paper, we studied level graphs of VGs and investigated HMs, WIs, and CWIs of VGs by HMs of level graphs. Likewise, we characterized some VGs by their level graphs.
2. Preliminaries
In this section, we review some concepts of graph theory and VGs.
Definition 1. Let be a finite nonempty set. A graph on consist of a vertex set and an edge set , where an edge is an unordered pair of distinct nodes of . We will use rather than to denote an edge. If is an edge, then we say that and are neighbor. A graph is called complete graph if each pair of nodes are neighbor.
Definition 2. Let and be graphs. A mapping is a homomorphism from to if and are neighbor whenever and are neighbor.
Definition 3. Two graphs and are isomorphic if there is a bijective mapping so that and are neighbor in if and only if and are neighbor in , is named isomorphism from to . An isomorphism from a graph to itself is named an automorphism of . The set of all automorphisms of forms a group, which is named the automorphism group of and shown by .
Definition 4. A VS is a pair on set X where and are taken as real valued functions which can be defined on so that , .
Definition 5. Let . We say that is contained in and write , if for any ,Let . For any , the orders and on are defined asIt is easy to see that, constitutes a complete lattice with maximum element and minimum element .
Definition 6. Let . For each , we define .
Then, is named -level set of . The set is called the support and is denoted by .
Let be a finite nonempty set. Denote by the set of all 2-element subsets of . A graph on is a pair where , and are named vertex set and edge set, respectively.
Definition 7. Let be a finite nonempty set, and . The triple is named a VG on , if for each ,If is a VG, then, it is easy to see that is a graph and it is called underlying graph of . The set of all VG on is denoted by . For given , in this study suppose that .
Definition 8. Let and be two VGs. Then,(1)A mapping is a homomorphism from to , if(i)(ii)(2)A mapping is a weak isomorphism from to , if is a BH from to and (3)A mapping is a coweak isomorphism from to , if is a BH from to and (4)An isomorphism from to is a bijective mapping so that(i)(ii)
Definition 9. VG is called strong vague graph (SVG) if , and is called complete vague graph (CVG), if , . A CVG with nodes is denoted by .
Definition 10. Suppose that and be two VGs. Then, is VSG of , if and .
Definition 11. Let be VG and . Then, the VG so that , , is named the induced VSG by and shown by .
Definition 12. A family of VSs on is named a -coloring of VG if(i).(ii) for .(iii)For each strong edge of , for . We say that a graph is -colorable if it can be colored with colors.All the basic notations are shown in Table 1.
3. Homomorphisms and Isomorphisms of Vague Graphs
In this section, we discuss the homomorphism and isomorphism of VGs by the homomorphism of level graphs in VGs.
Theorem 1. Let be a finite nonempty set, and . Then, if and only if is a graph for all ,.
Proof. Let be VG. For each , , assume that . Then, and . Because is VG,It follows that . Therefore, is a graph.
Conversely, let is a graph, , . For each , let Then, . Hence, . Thus, . This implies that and . Therefore, is VG.
Definition 13. Let and be two VGs, a mapping. For any , , if is a homomorphism from to , then, is called homomorphism mapping from to .
Theorem 2. Let and be two VGs. Then, is a homomorphism from to if and only if is - homomorphism from to.
Proof. Assume that is a homomorphism from to . Let, , . If , thenHence, implying is a mapping from to . For , let . Then,Hence,which implies . Therefore, is a homomorphism from to .
Conversely, let be a - homomorphism from to . For arbitrary element , let , . Then, , hence, , because is a homomorphism from to . It follows thatthat is,Now for arbitraries , let , . Then,Hence, and . Because is a homomorphism from to , we conclude that and . Therefore,
Theorem 3. Let and be two VGs. Then, is a WI from to if and only if is a bijective - homomorphism from toand
Proof. Let be a WI from to . From the definition of homomorphism is a bijective homomorphism from to . By Theorem 2 is a bijective - homomorphism from to and also by the definition of WI we haveConversely, from hypothesis, is a bijective mapping andFor , let , . Then,which implies and . Because is a homomorphism from to , we have and . Hence,which complete the proof.
Theorem 4. Let and be two VGs. Then, is a CWI from to if and only if is a bijective - homomorphism from to and
Proof. Let be a CWI from to . Then, is a bijective homomorphism from to . By Theorem 2 is a bijective - homomorphism from to . Also by the definition of CWIConversely, from hypothesis, we know that is a bijective mapping andFor arbitrary element , suppose that , . Then, we have . Now because is a homomorphism from to , . Thus, , , which implies is a CWI from to .
Corollary 1. Let , . Ifis a CWI fromto, then,is an IH fromto, , .
From the following example, we conclude that the converse of Corollary 1 do not need to be true.
Example 1. Let and be two VGs, as shown in Figure 1. Consider the mapping , defined by , . In view of the - level graphs of and in Figure 1, if then, is an IH from to , but is not a CWI.
Theorem 5. Let,, and be a mapping. For each , , if is an isomorphism from to a SG of, then,is a CWI fromto an induced VSG of.
Proof. The mapping is an isomorphism from to a SG , so is an IM. For arbitrary , suppose that , . Then, , and so . Hence, and . For , let and . Then, , , , and . Hence, and . Since isomorphism from to , we get and . Therefore,Now, let , . Then, . Because is injective and an isomorphism from to a SG of , we have and . Therefore,Now by (20) and (21), we conclude that
Corollary 2. Letand be two VGs with , and a mapping. For ,, if is an isomorphism from to a SG of, then,is a CWI fromto.
Theorem 6. Letandbe two VGs,be a bijective mapping. If for each,is an isomorphism fromto, then,is an isomorphism fromto.
Proof. From hypothesis, is a bijective mapping and an isomorphism from to . By Theorem 5 is a CWI from to and is a CWI from to . Therefore, is an isomorphism from to .
Corollary 3. Letbe VG anda bijective mapping. Then,is an automorphism ofif and only ifis an automorphism of, from an,.
Theorem 7. Let be VG. Then, is a CVG if and only if is a CG for.
Proof. If is a CVG and for , , , then, , , , , and soHence, . It follows that is a CG. Conversely, suppose that is not a CVG. Then, there are so that or . Let , and , for . Then, and . Hence, , for a , but . This implies that is not a CG. For the case , it follows similarly.
Theorem 8. Let. Then,has not IV, for each,if and only if for each,so that,.
Proof. Suppose that for each , , graph has not IV and there is a node so that for each , or . Let and , , for . Then, and for each , , . Therefore, is an IV in the graph , which is a contradiction.
Now suppose that for , , node is an IV in . If , then, or , and if , it is trivial that , hence, or . Therefore, for each , , .
Here, we describe the relationship between coloring graph and homomorphism of graph.
Theorem 9. A VG is -colorable there exists a homomorphism from to the .
Proof. Assume that be -colorable with colors labeled . Let . We define CVG with vertices set , so that the degree of membership vertex is and the degree of non-membership vertex is . Now the mapping defined by is a graph homomorphism, becauseAccording to the definition of CVG, for and we haveThen, , , for all .
Conversely, let be a homomorphism. For a given , define the set to beIf , let , otherwise . Therefore, the VG is -colorable with coloring set .
4. Application
Nowadays, the issue of coloring is very important in the theory of fuzzy graphs because it has many applications in controlling intercity traffic, coloring geographical maps, as well as finding areas with high population density. Therefore, in this section, we have tried to present an application of the coloring of vertices in a VG.
Example 2. Let be a VG (See Figure 2). We modeled a FG by considering countries as vertices of graph. The membership and nonmembership value of the vertices are the good and bad activity of a country with respect technology so that are , , , , respectively. There is an edge if they share a boundary. Let , , , , and are edges of graph . The membership and nonmembership value of the edges are the political relationship in a good and bad attitude such that , , , , , respectively. We now want to see how many days we will need to hold a conference between these countries. Let be a set of countries; and Suppose that be countries have boundary for . Now, form FG with vertices set , where are neighbor if and only if . For instance, and . So and hence , are neighbor. By Theorem 9 there is a homomorphism from to complete graph with . Then, 3 days are required to hold a conference between these countries, . The colored graph of the example 2 is shown in Figure 3.
In the next example, we want to identify the most effective employee of a hospital with the help of a vague influence digraph.
Example 3. Hospitals are very important organizations whose existence is directly related to the general health of the community. Researchers in each country examine factors that contribute to the success of strategic planning to improve the management status of these health organizations. The lives and health of many people are in the hands of health systems. From the safe delivery of a healthy baby to the respectful care of an elderly person, the health department has a vital and ongoing responsibility to individuals throughout their lives. The health industry has undergone many political, social, economic, environmental, and technological changes since the early 1980s. These changes have created challenges for managers of healthcare organizations, especially hospitals that cannot be managed with operational plans. Thus, hospital managers have resorted to strategic planning since the 1980s to achieve excellence. Since the management in each ward of the hospital is very important, so in this section, we have tried to determine the most effective person in a hospital based on the performance of its staff. Therefore, we consider the vertices of the VIG as the heads of each ward of the hospital, and the edges of the graph as the degree of interaction and influence of each other. For this hospital, the set of staff is .(i)Ameri has been working with Taleshi for 14 years and values his views on issues.(ii)Taheri has been responsible for audiovisual affairs for a long time, and not only Ameri, but also Taleshi, are very satisfied with Taheri’s performance.(iii)In a hospital, the preservation of medical records is a very important task. Kamali is the most suitable person for this responsibility.(iv)Talebi and Kamali have a long history of conflict.(v)Talebi has an important role in the radiology department of the laboratory.Given the abovementioned, we consider this a VIG. The vertices represent each of the hospital staff. Note that each staff member has the desired ability as well as shortcomings in the performance of their duties. Therefore, we use of VS to express the weight of the vertices. The true membership indicates the efficiency of the employee and the false membership shows the lack of management and shortcomings of each staff. But the edges describe the level of relationships and friendships between employees such that the true membership shows a friendly relationship between both employees and the false membership shows the degree of conflict between the two officials. Names of employees and levels of staff capability are shown in Tables 2 and 3. The adjacency matrix corresponding to Figure 4 is shown in Table 4.
Figure 4 shows that Najafi has of the power needed to do the hospital work as the medical equipment expert, but does not have the knowledge needed to be the boss. The directional edge of Taleshi–Ameri shows that there is friendship among these two employees, and unfortunately, they have conflict. Clearly, Badri has dominion over both Kamali and Najafi, and his dominance over both is . It is clear that Badri is the most influential employee of the hospital because he controls both the head of the medical equipment and the medical records archive expert, who have of the power in the hospital.
5. Conclusion
VGs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. With the help of VGs, the most efficient person in an organization can be identified according to the important factors that can be useful for an institution. Hence, in this paper, we introduced the notion of - homomorphism of VGs and classify HMs, WIs, and CWIs of VGs by - homomorphisms. We also investigated the level graphs of VGs to characterize some VGs. Finally, we presented two applications of VGs in coloring problem and also finding effective person in a hospital. In our future work, we will introduce new concepts of connectivity in VGs and investigate some of their properties. Also, we will study the new results of connected perfect dominating set, regular perfect dominating set, and independent perfect dominating set on VGs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Key R&D Program of China (Grant 2019YFA0706402) and the National Natural Science Foundation of China under Grant 62172302, 62072129, and 61876047.