A Survey on Domination in Vague Graphs with Application in Transferring Cancer Patients between Countries

: Many problems of practical interest can be modeled and solved by using fuzzy graph (FG) algorithms. In general, fuzzy graph theory has a wide range of application in various ﬁelds. Since indeterminate information is an essential real-life problem and is often uncertain, modeling these problems based on FG is highly demanding for an expert. A vague graph (VG) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems in which fuzzy graphs may not succeed in bringing about satisfactory results. Domination in FGs theory is one of the most widely used concepts in various sciences, including psychology, computer sciences, nervous systems, artiﬁcial intelligence, decision-making theory, etc. Many research studies today are trying to ﬁnd other applications for domination in their ﬁeld of interest. Hence, in this paper, we introduce different kinds of domination sets, such as the edge dominating set (EDS), the total edge dominating set (TEDS), the global dominating set (GDS), and the restrained dominating set (RDS), in product vague graphs (PVGs) and try to represent the properties of each by giving some examples. The relation between independent edge sets (IESs) and edge covering sets (ECSs) are established. Moreover, we derive the necessary and sufﬁcient conditions for an edge dominating set to be minimal and show when a dominance set can be a global dominance set. Finally, we try to explain the relationship between a restrained dominating set and a restrained independent set with an example. Today, we see that there are still diseases that can only be treated in certain countries because they require a long treatment period with special medical devices. One of these diseases is leukemia, which severely affects the immune system and the body’s defenses, making it impossible for the patient to continue living a normal life. Therefore, in this paper, using a dominating set, we try to categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in the countries better suited in terms of both cost and distance.


Introduction
Fuzzy set theory and the related fuzzy logic were proposed by Zadeh [1] for dealing with and solving various problems in which variables, parameters, and relations are only imprecisely known, and hence, for which approximate reasoning schemes should be used. Such a situation is common in the case of virtually all nontrivial and, in particular, humancentered phenomena, processes, and systems that prevail in reality, and it is difficult to use conventional mathematics, based on binary logic, for their adequate characterization. Fuzzy set theory has been developed in many directions and has evoked great interest among mathematicians and computer scientists working in different fields of mathematics. Rosenfeld [2] used the concept of a fuzzy subset of a set to introduce the notion of a fuzzy subgroup of a group. Rosenfeld's paper spearheaded the development of fuzzy abstract algebra. Zadeh [3] introduced the notion of interval-valued fuzzy sets as an arcs. The domination concept in intuitionistic fuzzy graphs was examined by Parvathi and Thamizhendhi [46]. Manjusha and Sunitha [47,48] studied coverings, matchings, and paired domination in fuzzy graphs using strong arcs. Gang et al. [49] investigated total efficient domination in fuzzy graphs. Karunambigai et al. [50] introduced domination in bipolar fuzzy graphs. Cockayne [51] and Hedetniemi [52] described the independent and irredundance domination number in graphs.
Domination in PVGs is so important that it can play a vital role in decision-making theory, which concerns finding the best possible state in a test or experiment. Although some DSs in FGs have been proposed by a number of authors, the breadth of the subject and its various applications in engineering and medicine, and the limitations of past definitions, prompted us to introduce new types of DSs on the PVG. In fact, the limitations of the old definitions in FGs forced us to come up with new definitions in PVGs. Hence, in this study, we represent different kinds of DSs, such as EDS, TEDS, GDS, and RDS, in PVGs and also try to described the properties of each by giving some examples. The relationship between IESs and ECSs are established. A comparative study between "EDS and Minimal-EDS", and "IES and Maximal-IES" was conducted. Some remarkable properties associated with these new DSs were investigated, and the necessary and sufficient condition for a DS to be a GDS was stated. Finally, we defined RIS and RDS and examined the relationships between them in a theorem. Today, many cancer patients pass away due to the lack of the necessary medical equipment in their country. Therefore, it is indispensable to identify the countries that have the necessary conditions to treat such patients. Hence, in this paper, with the help of DS, we attempt to identify countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in these countries, which are better suited in terms of both cost and distance.

Preliminaries
In the following, some basic concepts on VGs are reviewed in order to facilitate the next section.
A graph G * = (V, E) is a mathematical structure containing of a set of nodes V and a set of edges E, so that every edge is an unordered pair of distinct nodes. An FG has the form of ζ = (γ, ν), where γ : V → [0, 1] and ν : V × V → [0, 1] is defined as ν(mn) ≤ γ(m) ∧ γ(n), ∀m, n ∈ V, and ν is a symmetric fuzzy relation on γ, and ∧ denotes the minimum.

Definition 1. ([8])
A VS A is a pair (t A , f A ) on set V, where t A and f A are used as real-valued functions, which can be defined on V → [0, 1], so that t A (m) + f A (m) ≤ 1, ∀m ∈ V. G * is a crisp graph (V, E) and ζ a VG (A, B) throughout this paper.
By routine computations, it is easy to show that ζ is a VG.
called the order of a VG ζ, and is denoted by p(ζ); is called the size of a VG ζ, and is denoted by q(ζ).

Example 2.
In Example 1, it is easy to show that Definition 4. ( [46]) Let ζ = (A, B) be a VG. If m i , m j ∈ V, then the t-strength of connectedness between m i and m j is defined as t ∞ B (m i , m j ) = sup{t k B (m i , m j )|k = 1, 2, · · · , n} and the fstrength of connectedness between m i and m j is defined as f ∞ B (m i , m j ) = inf{ f k B (m i , m j )|k = 1, 2, · · · , n}. Furthermore, we have t k B (mn) = sup{t B (m, n 1 ) ∧ t B (n 1 , n 2 ) ∧ t B (n 2 , n 3 ) ∧ · · · ∧ t B (n k−1 , n)| (m, n 1 , n 2 , · · · , n k−1 , n) ∈ V} and f k   Clearly, e 1 and e 2 are strong edges.

Definition 6.
( [46]) The two vertices m i and m j are said to be neighbors in a VG ζ if either one of the following conditions holds: The two vertices m i and m j are said to be strong A subset S of V is called a DS in ζ if for each m ∈ V − S, ∃ n ∈ S, so that m dominated n. A DS S of a VG ζ is referred to as a minimal DS if no proper subset of S is a DS.

Example 5.
Consider the VG ζ as in Figure 3. It is easy to show that {m, n, z} and {n, z, w} are DSs.   Obviously, ζ is a PVG since it has the condition of Definition 10.

Definition 11. ([43]) An edge mn in a PVG
Example 7. In Example 6, mn is an effective edge.

Definition 12. ([43])
If ζ be a PVG, then the vertex cardinality of S ⊆ V is defined as Definition 13. ( [43]) Let ζ = (A, B) be a PVG, then the edge cardinality of K ⊆ E is defined as Definition 14. ( [43]) Two edges mn and zw in a PVG ζ are said to be adjacent if they are neighbors. Furthermore, they are independent if they are not adjacent.

Definition 15. ([29]) Let ζ = (A, B) be a PVG. A vertex subset S of V(ζ) is called a DS of ζ if
for every node m ∈ V − S, ∃ a node n ∈ S, so that Example 8. Consider the PVG ζ as in Figure 5. It is easy to show that S = {n, z} is a DS. Definition 17. ( [43]) Two nodes in a PVG ζ are said to be independent if there is no strong arc between them. A subset S of V is said to be an independent set if every two nodes of S are independent.
All the basic notations are shown in Table 1.

Edge Domination in PVGs
The maximum cardinality among all maximal IES in ζ is called the EIN and it is denoted by β 1 (ζ) or simply β 1 .
Proof. By definition, K is an independent set if and only if no two edges of K are adjacent, if and only if every edge of K is incident with at least one vertex of E − K, and if and only if E − K is an ECS of ζ.

Example 12.
Consider the PVG ζ as in Figure 7. It is easy to show that K = {e 1 , e 3 } is an independent set and E − K = {e 2 , e 4 } is an ECS.

Definition 21.
An edge e of a PVG ζ is said to be an isolated edge if no effective edges are incident with the vertices of e. Hence, an isolated edge does not dominate any other edge in ζ.

Definition 24.
Let ζ = (A, B) be a PVG and e i and e j be two adjacent edges of ζ. We say that e i dominates e j if e i is an effective edge. An edge subset K of E in a PVG ζ is called an EDS if, for each edge e i ∈ E − K, ∃ an effective edge e j ∈ K so that e j dominates e i . An EDS K of a PVG ζ is said to be a minimal EDS if for each edge e ∈ K, K − {e} is not an EDS. The minimum cardinality between all minimal EDSs is called an EDN of ζ and is described by γ (ζ) or simply γ . An EDS K of a PVG ζ is said to be independent if t B (mn Example 16. Consider the VG ζ as in Figure 9. (ii) ∃ an edge e ∈ E − K so that N(e) ∩ K = {e} and e is an effective edge.
Proof. Let K be a minimal EDS and e ∈ K. Then, K e = K − {e} is not an EDS and hence ∃ t ∈ E − K e , so that t is not dominated by any element of K e . If t = e, we obtain (i) and if t = e, we obtain (ii). Conversely, assume that K is an EDS, and for each edge e ∈ K, one of the two conditions holds. Suppose K is not a minimal EDS, then ∃ an edge e ∈ K, and K − {e} is an EDS. Therefore, e is a strong neighbor to at least one edge in K − {e}, and the first condition does not hold. If K − {e} is an EDS, then each edge in E − K is a strong neighbor to at least one edge in K − {e}, and the second condition does not hold, which contradicts our assumption that at least one of these conditions holds. Hence, K is a minimal EDS. Theorem 4. Let ζ = (A, B) be any PVG without isolated edges. Then, for each minimal EDS K, E − K is also an EDS.
Proof. Let e be any edge in K. Since ζ has no isolated edges, there is an edge t ∈ N(e). It follows from Theorem 3 that t ∈ E − K. Hence, each element of K is dominated by some element of E − K. Thus, E − K is an EDS in ζ.  Proof. Any PVG without isolated nodes has two disjoint EDSs and hence the result follows.

Theorem 6. An IES K of a PVG ζ is a maximal IES if and only if it is an IES and EDS.
Proof. Let K be a maximal IES in a PVG ζ and, hence, for each edge e ∈ E − K, the set K ∪ {e} is not independent. For each edge e ∈ E − K, ∃ an effective edge t ∈ K so that t dominates e. Hence, K is an EDS. Therefore, K is both an EDS and IES. Conversely, assume K is both independent and an EDS. Suppose that K is not a maximal IES, then ∃ an edge e ∈ E − K, and the set K ∪ {e} is independent. If K ∪ {e} is independent, then no effective edge in K is strong neighbor to e. Therefore, K cannot be an EDS, which is a contradiction. Thus, ζ is a maximal IES.

Example 19.
Consider the PVG ζ as in Figure 10. In Figure 10, {e 1 , e 2 , e 4 } is a minimal IES that is both an IES and EDS.

Theorem 7. Every maximal IES K in a PVG ζ is a minimal EDS.
Proof. Let K be a maximal IES in a PVG ζ. By Theorem 6, K is an EDS. Assume K is not a minimal EDS, ∃ at least one edge e ∈ K for which K − {e} is an EDS. However, if K − {e} dominates E − {K − {e}}, then at least one edge in K − {e} must be strong neighbor to e. This contradicts the fact that K is an IES in ζ. Hence, K must be a minimal EDS.

Definition 25.
Let ζ = (A, B) be a PVG without isolated edges. An edge subset K of E is said to be TEDS if for each edge e ∈ E, ∃ an edge t ∈ K, t = e, so that t dominates e.

Definition 27.
A DS K of a PVG ζ is called GDS if K is also a DS of ζ. The minimum cardinality among all GDSs is named GDN and is denoted by γ g (ζ).

Theorem 8. The GDS K in a PVG ζ is not a singleton.
Proof. The GDS K is a DS for both ζ and ζ and both of them are nonempty sets. Hence, K is not a singleton.

Example 22.
Consider the PVG ζ as in Figure 12. It is obvious that K 1 = {n, z} and K 2 = {m, w} are GDSs, which are also DSs in ζ, and neither are singletons.

Theorem 9.
A DS K is a GDS if and only if for every node n ∈ V − K, ∃ a node m ∈ K so that m and n are not dominating each other.
Proof. Let ζ be a PVG with a GDS K. Assume that m in K is dominating n in V − K, then K is not a DS, which contradicts the statement that K is a DS of ζ. Conversely, let for every n ∈ V − K, ∃ m ∈ K, so that m and n are not dominating each other, then K is a DS in both ζ and ζ, which gives that K is a GDS of ζ and so is the result.

Definition 28. Let ζ = (A, B) be a PVG. A subset K ⊆ V is called RDS if (i) each node in V − K is neighbor to some nodes in K;
(ii) all the nodes in K have the same degrees.

Theorem 10. An RIS is a maximal RIS of a PVG ζ if and only if it is an RIS and RDS.
Proof. Let K be a maximal RIS in a PVG ζ, then for each m ∈ V − K, the set K ∪ {m} is not an independent set, i.e., ∀m ∈ V − K, ∃ a node n ∈ K so that m is neighbor to n. Therefore, K is a DS of ζ and also an RIS of ζ. Therefore, K is an RIS and RDS.
Conversely, assume that K is both an RIS and RDS of ζ. We have to prove that K is a maximal RIS. Suppose that K is not a maximal independent set. Then, ∃ a node m / ∈ K so that K ∪ {m} is an independent set, there is no node in K neighbor to m, and hence, m is not dominated by K. Thus, K cannot be a DS of ζ, which is a contradiction. Therefore, K is a maximal RIS. Figure 14, {m, z} is a maximal RIS that is both RIS and RDS.

Application
Dominations are becoming increasingly significant as they can be applied in many areas, such as psychology, computer science, nervous systems, artificial intelligence, decisionmaking theory, etc. Many authors today are trying to find other uses for domination in their field of interest. Furthermore, domination sets provide system modelers with more freedom and is less restrictive in permissible membership grades. To fully understand the concept of dominating sets in vague graphs, we now present an important application of domination in a vague environment.

Domination in Cancer Patients and Their Transferability among Countries
Today, with the advancement of medical science, the mortality rates have decreased so much that, after a period of treatment in hospitals or private clinics, patients often continue their normal daily lives. Unfortunately, there are still diseases that can only be treated in certain countries because they require a long treatment period using special medical devices. One of these diseases is Leukemia, which severely affects the body's immune and defense systems and often patients do fully recover. Many poor countries are unable to cope with this disease, and many people die each year in these countries as a result of a lack of medical equipment. Therefore, in this paper, using dominating sets, we categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in these countries, which are better suited in terms of both cost and distance. For this purpose, we consider five countries: China, India, Indonesia, Taiwan, and Korea. We utilized the following website: https://www.wcrf.org/global-cancer-data-by-country/, which we accessed on 12 April 2021. This website modeled the number of cancer patients in different countries as a vague graph. Table 2 shows the number of cancer patients in these five countries in 2018 (according to the aforementioned website). Unfortunately, it is clear that many people suffer from this disease. Moreover, most countries do not have the adequate medical facilities to diagnose and treat the disease. Table 3 indicates the amount of medical equipment in these countries by percentage (according to the global atlas of medical devices-World Health Organization, 2018). The amount of scientific literature on social inequality in health has increased exponentially in recent years. However, the effect of politics and policies on health and social inequality in health is rarely a focus. This is a schematic attempt to show how politics is related to the expansion of the welfare state, in turn reflecting the degree to which societies take care of their citizens. The welfare state and labour market policies have an effect on the income and social inequality in the population. Obviously, countries with better political relations are also better able to solve their medical problems. This is clearly shown in [53].
Suppose that there is a cancer patient in Indonesia who wants to travel to one of the four countries for treatment. For our patient, the conditions and reasons for transferring from the country of origin to another country for treatment include the following: Firstly, the patient's financial situation and social level are of importance in order to meet the costs of treatment in another country. Secondly, the scientific level of the destination country and the existence of specialized clinics and hospitals must be adequate to ensure that comprehensive and centralized treatment options are available in one center, including radiotherapy, immunotherapy, bone marrow transplant, etc. Note that, in Figure 15, we consider the conditions for transporting a patient to the destination country as "facilities" and the medical facilities of the destination country as "equipment". In this vague graph, the nodes represent the countries and the edges represent the extent of the political relations between the countries. The weights of the vertices and edges are given in Tables 4 and 5.
It is obvious that K 1 has the smallest size as compared with the other DSs; hence, we concluded that it is the best choice. This is because, firstly, China has more powerful medical equipment than other countries; and secondly, there is a stronger friendly relationship between China and Indonesia. As we can see, despite the fact that China has the highest number of patients among these five countries, its hospitals and clinics are equipped with powerful diagnostic and therapeutic tools for the treatment of this disease. Furthermore, the high number of patients is related to the population size of this country and its superior diagnostic facilities. Therefore, governments should try to reduce their political differences so that patients can easily seek treatment in other powerful countries.

Comparison with Distance between Countries
In this subsection, we intend to examine another influential factor, i.e., the distance or distance between countries. This can play a significant role in deciding which country to choose for treatment when using the dominating set. The distances between Indonesia and other countries are presented in Table 6.
Information about the distances between countries was obtained from the following website (https://www.geodatos.net/en/distances/countries, accessed on 12 April 2021). According to Figure 15, the minimum edge dominating sets are as follows: After calculating the cardinality of K 1 , K 2 , K 3 , and K 4 , we have It is clear that K 1 has the smallest size as compared with the other edge dominating sets. Therefore, we concluded that it is the most appropriate choice as compared to the other edge dominating sets. Moreover, according to Table 6, it is clear that Indonesia-Taiwan has the shortest distance between them. Therefore, a comparison between these two subsections shows that the dominating sets always provide the best possible condition for the treatment of the patient.

Conclusions
A vague model is suitable for modeling problems with uncertainty, indeterminacy, and inconsistent information in which human knowledge is necessary and human evaluation is required. Vague models give more precision, flexibility, and compatibility to the system as compared to classical, fuzzy, and intuitionistic fuzzy models. A vague graph can describe the uncertainty of all kinds of networks well. The VG concept has a wide variety of applications in different areas, such as computer sciences, operation research, topology, and natural networks. Moreover, the term domination has a wide range of applications in graph theory for the analysis of vague information. Domination in FGs theory is one of the most widely used topics in various sciences, including psychology, computer science, nervous systems, artificial intelligence, etc. Hence, in this research, we describe different kinds of DSs, such as EDS, TEDS, GDS, and RDS, in PVGs. Furthermore, we present the properties of each by giving various examples, and the relationship between IESs and ECSs are established. Moreover, we derived the necessary and sufficient condition in which an edge dominating set to be minimal. We also show when a dominance set can be a global dominance set. Finally, we introduce an application of domination in the field of medical science. In future work, we will introduce a vague competition graph and study new types of domination, such as regular perfect DS, inverse perfect DS, equitable DS, and independent DS on vague competition graphs.
Author Contributions: Y.R. and S.K. conceived and designed the experiments; R.C. performed the experiments; P.W. and R.C. analyzed the data; H.J. and S.K. contributed reagents/materials/analysis tools; Y.R. wrote the paper. All authors have read and agreed to the published version of the manuscript.