A Study on Domination in Vague Incidence Graph and Its Application in Medical Sciences

: Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been acknowledged as being an applicable and well-organized tool to epitomize and solve many multifarious real-world problems in which vague data and information are essential. Owing to unpredictable and unspeciﬁed information being an integral component in real-life problems that are often uncertain, it is highly challenging for an expert to illustrate those problems through a fuzzy graph. Therefore, resolving the uncertainty accompanying the unpredictable and unspeciﬁed information of any real-world problem can be done by applying a vague incidence graph (VIG), based on which the FGs may not engender satisfactory results. Similarly, VIGs are outstandingly practical tools for analyzing different computer science domains such as networking, clustering, and also other issues such as medical sciences, and trafﬁc planning. Dominating sets (DSs) enjoy practical interest in several areas. In wireless networking, DSs are being used to ﬁnd efﬁcient routes with ad-hoc mobile networks. They have also been employed in document summarization, and in secure systems designs for electrical grids; consequently, in this paper, we extend the concept of the FIG to the VIG, and show some of its important properties. In particular, we discuss the well-known problems of vague incidence dominating set, valid degree, isolated vertex, vague incidence irredundant set and their cardinalities related to the dominating, etc. Finally, a DS application for VIG to properly manage the COVID-19 testing facility is introduced.


Introduction
The graph concept stands as one of the most dominant and widely employed tools for the multiple real-world problem representation, modeling, and analyses. To represent the objects and the relations between them, the graph vertices and edges are applied, respectively. FG-models are beneficial mathematical tools for addressing the combinatorial problems in various fields involving research, optimization, algebra, computing, environmental science, and topology. Thanks to the natural existence of vagueness and ambiguity, fuzzy graphical models are strikingly better than graphical models. Originally, fuzzy set theory was required to contend with many multidimensional issues, which are replete with incomplete information. In 1965 [1], fuzzy set theory was first proposed by Zadeh. Fuzzy set theory is a highly influential mathematical tool for solving approximate reasoning related problems. By presenting the VS notion through changing the value of an element in a set with a sub-interval of [0, 1], Gau and Buehrer [2] introduced and structured the vague set theory. The VSs describe more possibilities than fuzzy sets. A VS is more effective for the existence of the false membership degree. An immediate result of a rise of popularity of fuzzy sets theory has been the fuzzification of graph theory which has been initiated by Rosenfeld [3] who has introduced the concept of a fuzzy graph. The incidence graphs can generally be represented as a triple (V, E, I), where V is a finite set of vertices and E is a finite set of edges, and I ⊆ V × E is an incidence function which indicates for each edge its end vertices and whether the edge is directed (1) or not (0). If an edge e is directed, then the first element and the second element of I(e) denote the origin vertex and the destination vertex, respectively.
Domination in graphs has many applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospital, fire stations) is fixed and one attempts to minimize the distance that a person needs to travel to the closest facility. Concepts from DS also appear in problems involving finding sets of representative in monitoring communication or electrical networks. The concept of a domination in an FG was introduced by Somasundaram [32]. Parvathi and Thamizhendhi [33] described the concept of a domination number in an intuitionistic fuzzy graph. Gani et al. [34] gave some properties of a fuzzy-DS, and a fuzzy irredundant set. Many researchers, notably Talebi and Rashmanlou [35], studied new applications of the concept of domination in VGs. The vague incidence model is more compliant and functioning than fuzzy and intuitionistic incidence fuzzy models. In several applications, such as urban traffic planning, telecommunication message routing, very large-scale integration chip optimal pipelining, etc., VIGs serve as the observed real-world systems mathematical models. Therefore, in this research, we extend the concept of the FIG to the VIG and discuss the well-known problems of vague incidence dominating set (VIDS), valid degree, vague incidence irredundant set (VIIS), and their cardinalities related to the domination, etc.
Many people today suffer from COVID-19 virus and in some cases die. Unfortunately, the lack of necessary facilities in hospitals and clinics has multiplied the possibility of the virus outbreak. Hence, an application of DS for VIG to properly manage the COVID-19 testing facility is introduced.

Preliminaries
In this section, to set the stage for our analysis, and to facilitate the following of our discussion, we show a brief overview of some of the basic definitions. A graph is a pair of G = (V, E) that holds E ⊆ V × V. The elements of V(G) and E(G) are the nodes and edges of the graph G, respectively. An FG is of the form G = (ψ; φ) which is a pair of mapping ψ : V → [0, 1] and φ : V × V → [0, 1] as is defined as φ(mn) ≤ ψ(m) ∧ ψ(n), ∀m, n ∈ V, and φ is a symmetric fuzzy relation on ψ and ∧ denotes the minimum. Definition 1. [4] Let G = (V, E) be a graph. Then, G * = (V, E, I) is called an incidence graph, so that I ⊆ V × E. If V = {m, n}, E = {mn} and I = {(m, mn)}, then (V, E, I) is an incidence graph even though (n, mn) / ∈ I. The pair (m, mn) is called an incidence pair or simply a pair. If (m, mn), (n, mn), (n, nz), (z, nz) ∈ I, then mn or nz are called adjacent edges. Definition 2. [6] Let G * = (V, E, I) be an incidence graph and σ be a fuzzy subset of V and µ, a fuzzy subset of E. Let ψ be a fuzzy subset of I. If ψ(v, mn) ≤ σ(v) ∧ µ(mn), for all v ∈ V and mn ∈ E, then, ψ is called a fuzzy incidence of graph G * and G = (σ, µ, ψ) is called a FIG of G * .  By routine computations, it is easy to show that G is a VG. Definition 5. [21] The strength of a vague edge mn of a VG G = (A, B) is given as: .

Definition 6. [21]
A vague edge mn of a VG G = (A, B) is said to be vague strong edge whenever M mn ≥ 0.5 or N mn ≤ 0.5, otherwise is vague weak edge.
All the basic notations are shown in Table 1.

Vague Incidence Graph
A natural follow-up to the concept of a VG, outlined in the previous section, is the concept of a VIG which will now be presented with its main properties. Also, we will discuss a very important concept of domination on VIG.

Definition 10.
If ζ = (A, B, C) is a VIG, then, a vague incidence edge mn of ζ is called a vague incidence valid and otherwise it is called a vague incidence invalid edge.

Definition 11.
If ζ is a VIG, then, its cardinality is defined as: and also, we have: Example 3. Consider a VIG, ζ as shown in Figure 4. Here, the edges of mt, tw, and wz are the vague incidence valid edges.
Example 4. In Figure 4, it is obvious that mn and nk are vague incidence valid edges and we have: ∀v ∈ V and mn ∈ E.
Example 5. Consider a VIG, ζ as shown in Figure 5. It is easy to see that ζ is a complete VIG.

Definition 14.
The vague incidence valid neighborhood degree of vertex m is defined as: and the minimum and maximum cardinality vague incidence valid neighborhood degrees of ζ are denoted by δ N and ∆ N , respectively.

Definition 15. If ζ = (A, B, C) is a VIG, then, the vertex cardinality of S ⊆ V is defined as
Example 7. Consider the VIG ζ as shown in Figure 4. We have: Hence, it is clear that δ N = (0.1, 0.5) and ∆ N = (0.5, 0.8).

Definition 16.
If ζ = (A, B, C) is a VIG of G * , then, we say that m ∈ V incidentally dominates n ∈ V in ζ whenever n ∈ N v [m].

Definition 17.
A non-empty set of vertices D ⊆ V is called a VIDS in a VIG ζ whenever there exists n ∈ D so that m incidentally dominates n, i.e., V = N v [D], for all m ∈ V − D.

Definition 18. (i)
The minimum cardinality of VIDSs in a VIG ζ is called a vague incidence domination number of ζ and it is denoted by γ iv (ζ); clearly γ iv (ζ) ≤ p.
(ii) A set with the minimum vague cardinality equal γ iv (ζ), is called an γ iv -set or a minimal VIDS.

Definition 19.
A VIDS D of a VIG ζ is called a minimal VIDS whenever any proper subset of D is not a VIDS of ζ.
The upper vague incidence domination number Γ iv equals to the maximum vague cardinality of the minimal VIDSs in ζ. Clearly, γ iv is minimum vague cardinality of the minimal VIDS in ζ.
Proof. If D is a minimal VIDS of a VIG ζ, then Hence, γ iv ≥ p 1 + ∆ N .

Definition 21.
In a VIG ζ = (A, B, C), a vertex m ∈ V is called an isolated vertex if N v (m) = ∅, i.e., for any n ∈ V where n = m, mn is not a vague incidence valid edge.
Example 11. In Figure 4 we see that n is an isolated vertex because N v (n) = ∅.
Clearly, if m ∈ N vp (m, S), then, m is isolated in S .

Definition 25. (i)
The minimum cardinality among all maximal-VIISs, is called a vague incidence irredundance number ant it is denoted by ir v (ζ).
(ii) The maximum cardinality among all maximal-VIISs, is called an upper vague incidence irredundance number ant it is denoted by Ir v (ζ). It is clear that ir v (ζ) ⊆ Ir v (ζ).
Example 13. In Figure 7, it is easy to see that S = {m, z, t} is the only maximal VIIS and we have Ir v (ζ) = 1.25.
Proof. Let S be VIIS in ζ and m ∈ S. Assume that m is a valid neighborhood to k vertices in S. Since the degree of m is at least δ N , m must be valid neighborhood to at least δ N − dN k vertices in V − S where dN k is the cardinality of k valid neighborhood vertices m in S. If dN k = 0, then, |V − S| ≥ δ N , i.e., |S| ≤ p − δ N , as required. If dN k > 0, then, each valid neighbor of m in S must have a valid private neighborhood in V − S and these k vertices must be distinct. Hence,

Theorem 5. A VIDS in a VIG ζ = (A, B, C) is a minimum VIDS if and only if it is VIIS.
Proof. Let S be a minimal VIDS in ζ. Then, for any vertex m ∈ S, there exists a vertex z ∈ V − (S − {n}) which is not dominated by S − {n}. Therefore, for any n ∈ S, there holds N vp [m, S] = ∅. Thus, any vertex n ∈ S, has at least one incidence valid private neighborhood. Hence, S is both a VDS and a VIDS. Suppose that set S is not a minimal VIDS. Then, there exists an n ∈ S such that S − n is a VDS. Since S is a VIIS, then we obtain that N vp [n, S] = ∅. Let k ∈ N vp [n, S]. Then, w is not a valid neighborhood for any vertex in S − {n}, i.e., S − {n} is not a VDS which is a contradiction.

Theorem 6. Each minimal VIDS in a VIG is a maximal VIIS.
Proof. Due to Theorem 5, any minimal VIDS is a VIIS. Therefore, we just need to show that S is a minimal VIIS. If S is not a minimal VIDS, then, there exists a vertex m ∈ V − S such that S ∪ {m} is a VIIS. This means, N vp [m, S ∪ {m}] = ∅, so there exists at least one vertex n that is a valid private neighborhood of m into S ∪ {m}. Hence, no vertex in S is a neighborhood to n. This is a contradiction to S being a DS.

Application of VIDS for COVID-19 Testing Facility
Many people today have been infected with Covid-19 virus and in some cases died. Unfortunately, the necessary facilities deficiency in hospitals and clinics has increased the possibility of spreading the virus. Therefore, in this section, we have tried to identify the most effective medical centers with the help of the VIDS to save both cost and time. For this purpose, suppose that there are six different medical diagnostic clinics which are working in a city for conducting COVID-19 tests. Consider the clinics c 1 , c 2 , c 3 , c 4 , c 5 , and c 6 . In VIGs, the vertices show the clinics and edges show the contract conditions between the clinics to share the facilities or test kits. The incidence pairs show the transferring of patients from one clinic to another because of the lack of resources such as kits and doctors. The graph VIDS is the set of clinics which perform the tests independently.
The vertex c 3 (0.3, 0.5) means that it has 30% of the necessary facilities for diagnosis and treatment of the disease and unfortunately lacks 50% of the necessary equipment. The edge c 2 c 3 (0.2, 0.8) shows that there is only 20% of the interaction and relationship between the two clinics, and due to financial issues and competition, there is 80% on the conflict between them. The dominating sets for the Figure 8 are as follows: It is clear that D 1 has the smallest size among other dominating sets, so we conclude that it can be the best choice because, first, there are fewer patients who have the opportunity to transmit the disease to others, and secondly, it will save time and money. Therefore, the government should provide more facilities in the case of COVID-19 disease diagnosis as well as its treatment to clinics so that patients can be identified as soon as possible and do not have to go to different clinics to transmit the disease and spend a lot of money.

Conclusions
VIGs are highly practical tools for the study of different computational intelligence and computer science domains. VIGs have many applications in different sciences such as topology, natural networks, and operation research. DSs are a practical interest in several areas. In wireless networking, DSs are used to find efficient routes within ad-hoc mobile networks. They have also been used in document summarization, and in designing secure systems for electrical grids. Therefore, in this paper, we introduced new concepts on VIGs, namely VIDS, isolated vertex, vague incidence irredundant set, and their cardinality related to the domination, etc. At the end, an application of DS for VIG to properly manage the COVID-19 testing facility was presented. In our future work we will investigate the concepts of Energy, spectrum, density, adjacency matrix, degree matrix, Laplacian matrix, and new operations on VIGs and give some applications of spectrum and Energy in VIGs and other sciences.