Novel Concepts in Vague Graphs with Application in Hospital’s Management System

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 ( η , c ) - HMs of VGs and classify homomorphisms (HMs), weak isomorphisms (WIs), and coweak isomorphisms (CWIs) of VGs by ( η , c ) - 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.


Introduction
Graphs, from ancient times to the present day, have played a very important role in various elds, 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. ese 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 di erent disciplines including logic, topology, algebra, analysis, information theory, arti cial intelligence, operations research, and neural networks and planning [2][3][4][5][6]. e FS focuses on the membership degree of an object in a particular set. But membership alone could not solve the complexities in di erent cases, so the need for a degree of membership was felt. To solve this problem, Gau and Buehrer [7] introduced false-membership degrees and dened a VS as the sum of degrees not greater than 1. e rst de nition of FGs was proposed by Kafmann [8] in 1993, from Zade's fuzzy relations [9,10]. But Rosenfeld [11] introduced another elaborated de nition 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][14][15][16] dened 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]. e 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] de ned vertex covering in cubic graphs. Krishna et al. [26] presented a new concept in cubic graphs. Rao et al. [27][28][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. us, 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.

Preliminaries
In this section, we review some concepts of graph theory and VGs.
Definition 1. Let V be a finite nonempty set. A graph G � (V, E) on V consist of a vertex set V and an edge set E, where an edge is an unordered pair of distinct nodes of G. We will use pq rather than p, q to denote an edge. If pq is an edge, then we say that p and q are neighbor. A graph is called complete graph if each pair of nodes are neighbor.
and h(q) are neighbor whenever p and q are neighbor.
Definition 3. Two graphs G 1 and G 2 are isomorphic if there is a bijective mapping φ: V 1 ⟶ V 2 so that p and q are neighbor in G 1 if and only if φ(p) and φ(q) are neighbor in G 2 , φ is named isomorphism from G 1 to G 2 . An isomorphism from a graph G to itself is named an automorphism of G. e set of all automorphisms of G forms a group, which is named the automorphism group of G and shown by Aut(G).
where t A and f A are taken as real valued functions which can be defined Definition 5. Let A, B ∈ VS(V). We say that A is contained in B and write A⊆B, if for any p ∈ V, Let K * � (η, c)|η, c ∈ [0, 1], η + c ≤ 1 . For any (η 1 , c 1 ), (η 2 , c 2 ) ∈ L * , the orders ≤ and < on K * are defined as (2) It is easy to see that, (K * , ≤ ) constitutes a complete lattice with maximum element (1, 0) and minimum element (0, 1).
If X � (V, A, B) is a VG, then, it is easy to see that X * � (A * , B * ) is a graph and it is called underlying graph of X.
e set of all VG on V is denoted by VG(V). For given X � (V, A, B) ∈ VG(V), in this study suppose that A * � V.
(4) An isomorphism from X 1 to X 2 is a bijective mapping φ: Journal of Mathematics .
We say that a graph is k-colorable if it can be colored with k colors.
All the basic notations are shown in Table 1.

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 V be a finite nonempty set, A ∈ VS(V) and
Hence, Hence, p, q ∈ A (e,t) and pq ∈ B (e,t) . Because h is a homomorphism from Proof. Let h be a WI from X to Y. From the definition of homomorphism h is a bijective homomorphism from X to Y. By eorem 2 h is a bijective (η, c)homomorphism from X to Y and also by the definition of WI we have Conversely, from hypothesis, h: For p, q ∈ V, let t B (pq) � e, f B (pq) � t. en, which implies p, q ∈ A (e,t) and pq ∈ B (e,t) . Because h is a homomorphism from (A (e,t) , B (e,t) ) to (A (e,t) ′ , B (e,t) ′ ), we have h(p), h(q) ∈ A (e,t) ′ and h(p)h(q) ∈ B (e,t) ′ . Hence, which complete the proof.
Proof. Let h: V ⟶ W be a CWI from X to Y. en, h is a bijective homomorphism from X to Y. By eorem 2 h is a bijective (η, c)homomorphism from X to Y. Also by the definition of CWI Conversely, from hypothesis, we know that h: For arbitrary element p ∈ V, suppose that t A (p) � c, f A (p) � d. en, we have p ∈ A (c,d) . Now because h is a homomorphism from (A (c,d) , From the following example, we conclude that the converse of Corollary 1 do not need to be true. A, B) and Y � (W, A ′ , B ′ ) be two VGs, as shown in Figure 1. Consider the mapping h: V ⟶ W, defined by h(v i ) � w i , 1 ≤ i ≤ 5. In view of the (η, c)level graphs of X and Y in Figure 1, if A (η,c) ≠ ∅ then, h is an IH from X (η,c) to Y (η,c) , but h is not a CWI. For each (η, c) ∈ K * , A (η,c) ≠ ∅, if h is an isomorphism from X (η,c) to a SG of Y (η,c) , then, h is a CWI from X to an induced VSG of Y.

Proof.
e mapping h is an isomorphism from , c ≥ f A (q) and pq ∈ B (η,c) . Hence, p, q ∈ A (η,c) and pq ∈ B (η,c) . Since h isomorphism from X (η,c) to Y (η,c) , we get h(p), h(q) ∈ A (η,c) ′ and h(p)h(q) ∈ B (η,c) ′ . erefore, Now, let t B′ (h(p)h(q)) � k, f B′ (h(p)h(q)) � s. en, h(p)h(q) ∈ A (k,s) ′ . Because h is injective and an isomorphism from X (k,s) to a SG of Y (k,s) , we have p, q ∈ A (k,s) and pq ∈ B (k,s) . erefore, Now by (20) and (21), we conclude that Proof. From hypothesis, h − 1 : W ⟶ V is a bijective mapping and an isomorphism from Y (η,c) to X (η,c) . By eorem 5 h is a CWI from X to Y and h − 1 is a CWI from Y to X. erefore, h is an isomorphism from X to Y. Proof. If X � (V, A, B) is a CVG and for (η, c) ∈ K * , Hence, pq ∈ B (η,c) . It follows that X (η,c) is a CG. Conversely, suppose that X � (V, A, B) is not a CVG.
Proof. Assume that X be r-colorable with r colors labeled According to the definition of CVG, for u ∈ V i and v ∈ V j we have Conversely, let g: X ⟶ K r,A′ be a homomorphism. For a given k ∈ V(K r,A ′ ), define the set h − 1 (k)⊆V to be If otherwise λ k (v) � 0. erefore, the VG X is r-colorable with coloring set λ 1 , λ 2 , . . . , λ r .

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. erefore, in this section, we have tried to present an application of the coloring of vertices in a VG. A 1 , B 1 ) be a VG (See Figure 2). We modeled a FG by considering countries A, B, C, D as vertices of graph. e membership and nonmembership value of the vertices are the good and bad activity of a country with respect technology so that are respectively. We now want to see how many days we will need to hold a conference between these countries. Let there is a homomorphism from G to complete graph with n � 3. en, 3 days are required to hold a conference between these countries, e 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. e 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. e health industry has undergone many political, social, economic, environmental, and technological changes since the early 1980s.
ese changes have created challenges for managers of healthcare organizations, especially hospitals that cannot be managed with operational plans. us, 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 e ective person in a hospital based on the performance of its sta . erefore, 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 in uence of each other. For this hospital, the set of sta is F Taheri, Ameri, Talebi, Taleshi, Najafi, Kamali, Badri .
(i) Ameri has been working with Taleshi for 14 years and values his views on issues. (ii) Taheri has been responsible for audiovisual a airs for a long time, and not only Ameri, but also Taleshi, are very satis ed 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 con ict.
(v) Talebi has an important role in the radiology department of the laboratory.
Given the abovementioned, we consider this a VIG. e vertices represent each of the hospital sta . Note that each sta member has the desired ability as well as shortcomings in the performance of their duties. erefore, we use of VS to express the weight of the vertices. e true membership indicates the e ciency of the employee and the false membership shows the lack of management and shortcomings of each sta . 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 con ict between the two o cials. Names of employees and levels of sta capability are shown in Tables 2 and 3. e adjacency matrix corresponding to Figure 4 is shown in Table 4. Figure 4 shows that Naja has 90% of the power needed to do the hospital work as the medical equipment expert, but does not have the 10% knowledge needed to be the boss. e directional edge of Taleshi-Ameri shows that there is 30% friendship among these two employees, and unfortunately, they have 40% con ict. Clearly, Badri has dominion over both Kamali and Naja , and his dominance over both is 60%. It is clear that Badri is the most in uential employee of the hospital because he controls both the head of the medical equipment and the medical records archive expert, who have 90% of the power in the hospital.

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 (η, c)homomorphism of VGs and classify HMs, WIs, and CWIs of VGs by (η, c)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
e authors declare that they have no conflicts of interest.