A Study on Regular Domination in Vague Graphs with Application

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Introduction
Zadeh [1] introduced the subject of a fuzzy set (FS) in 1995. Rosenfeld [2] proposed the subject of FGs. The definitions of FGs from the Zadeh fuzzy relations in 1973 were presented by Kaufmann [3]. Akram et al. [4][5][6] introduced several concepts in FGs. Irregular VGs, domination in Pythagorean FGs, and 2-domination in VGs were studied by Banitalebi et al. [7][8][9]. Gau and Buehrer [10] introduced the notion of a vague set (VS) in 1993. The concept of VGs was defined by Ramakrishna [11]. Akram et al. [12] introduced vague hypergraphs. Rashmanlou et al. [13][14][15] investigated different subjects of VGs. Moreover, Akram et al. [16][17][18] developed several results on VGs. Kosari et al. [19] defined VG structure and studied its properties. The concepts of degree, order, and size were developed by Gani and Begum [20]. Borzooei and Rashmanlou [21] proposed the degree of vertices in VGs. Manjusha and Sunitha [22] studied the paired domination. Haynes et al. [23] expressed the fundamentals of domination in graphs. Nagoor Gani and Prasanna Devi [24] suggested the reduction in the domination number of an FG and the notion of 2-domination in FGs [25] as the extension of 2-domination in crisp graphs. The domination number and the independence number were introduced by Cockayne and Hedetniem [26]. In another study, A. Somasundaram and S. Somasundaram [27] proposed the notion of domination in FGs. Kosari et al. [28] studied new concepts in intuitionistic FG with an application in water supplier systems.
Parvathi and Thamizhendhi [29] introduced the domination in intuitionistic FGs. Domination in product FGs and intuitionistic FGs was studied by Mahioub [30,31]. Karunambigai et al. [32] introduced the domination in bipolar FGs. Rao et al. [33][34][35] expressed certain properties of domination in vague incidence graphs. Shi and Kosari [36,37] studied the domination of product VGs with an application in transportation. The concept of DS in FGs, both theoretically and practically, is very valuable. A DS in FGs is used for solving problems of different branches in applied sciences such as location problems. In this way, the study of new concepts such as DS is essential in FG. Domination in VGs has applications in several fields. Domination emerges in the facility location problems, where the number of facilities is fixed and one endeavors to minimize the distance that a person needs to travel to get to the closest facility. Qiang et al. [38] defined the novel concepts of domination in VGs. The notions of total domination, strong domination, and connected domination in FGs using strong arcs were studied by Manjusha and Sunitha [39][40][41]. Cockayne et al. [42] and Haynes et al. [43] investigated the independent and irredundance domination numbers in graphs. Natarajan and Ayyaswamy [44] introduced the notion of 2-strong (weak) domination in FGs. New results of irregular intuitionistic fuzzy graphs were presented by Talebi et al. [45,46]. Talebi and Rashmanlou,in [47], presented the concepts of DSs in VFGs. Narayanan and Murugesan [48] expressed the regular domination in intuitionistic fuzzy graph. A few researchers studied other domination variations which are based on the above definitions such as independent domination [49], complementary nil domination [50], and efficient domination [51]. In this paper, we introduced a new notion of ððϵ 1 , ϵ 2 Þ, 2Þ-Regular DS in VG. Finally, an application is given.

Preliminaries
In this section, we present some preliminary results which will be used throughout the paper. Definition 1. A graph G * is a pair ðX, EÞ, where X is called the vertex set and E ⊆ X × X is called the edge set. Definition 2. A pair C = ðψ, ζÞ is an FG on a graph G * = ðX, EÞ, where ψ is an FS on X and ζ is an FS on E, such that for all sv ∈ E: Definition 3 (see [10] Definition 4 (see [11]). A pair C = ðM, ZÞ is called a VG on for all s, v ∈ X. Note that Z is called vague relation on M. A VG G is named strong if for all sv ∈ E: Definition 5. Assume C = ðM, ZÞ is a VG on G * , the degree of a vertex s is denoted as dðvÞ = ðd t ðsÞ, d f ðsÞÞ, where The order of C is defined as (i) The vertex cardinality of C is defined by (ii) The edge cardinality of C is defined by Definition 9. Let C = ðM, ZÞ be a VG on G * . The vertex cardinality of S ⊆ X of VG on G * is defined by Definition 10. Let C = ðM, ZÞ be a VG on G * . The neighborhood of a vertex s ∈ X is defined by The neighborhood degree (ND) is denoted by d N ðsÞ and defined by The minimum ND is δ N ðCÞ = ∧fd N ðsÞ: s ∈ Xg:

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The maximum ND is Δ N ðCÞ = ∨fd N ðsÞ: s ∈ Xg: Definition 11 (see [48]). Let C = ðM, ZÞ be a VG on G * . Suppose s and v are any two vertices in C. Then, v is called to Definition 12 (see [48]). Let C = ðM, ZÞ be a VG on G * .
In Table 1, we show the essential notations.
Example 1. Consider a VG on G * .
Definition 14. Let C = ðM, ZÞ be a VG on G * . A set S ⊆ X is called to be ððϵ 1 , ϵ 2 Þ, 2Þ-Regular vague strong dominating set (VSDS) if every vertex s in V − S is strongly dominated by two vertices of S and each vertex in S has degree ðϵ 1 , ϵ 2 Þ: The minimum vague cardinality of ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VSDS is named ððϵ 1 , ϵ 2 Þ, 2Þ-Regular vague strong domination number and denoted by η rvs ðCÞ.
Example 2. Consider a VG on G * .

Theorem 28.
A ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS C of a VG C is minimal if and only if for each s ∈ S, one of the following two conditions holds: Proof. Suppose S is a minimal ððϵ 1 , ϵ 2 Þ, 2Þ-Regular dominating set. Then, for every vertex s ∈ S, S − fsg is not a ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS. This means that some vertex v ∈ X − ðS − fsgÞ is not dominated by two vertices in S − fsg.
Since v is not dominated by S − fsg, v is dominated by two vertices of s and t of S. Then, the vertex v is adjacent to s, t in S. Therefore, NðvÞ ∩ S = fs, tg. Conversely, let S be ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS, and for every s ∈ S, one of the two conditions holds. Suppose S is not a minimal dominating set. Then, there exists s ∈ S such that S − fsg is ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS. Hence, s is dominated by at least two vertices in S − fsg. Therefore, condition (i) does not   Proof. Suppose S is maximal ððϵ 1 , ϵ 2 Þ, 2Þ-Regular independent VDS. It is trivial that S is ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS and S is independent. Conversely, consider that S is independent and ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS. Suppose S is not maximal; then, there exists v ∈ X − S such that S ∪ fvg is independent. Then, v ∈ X − S is not adjacent to any vertex in S which is a contradiction. Therefore, S is maximal.
Proof. Since each minimum ððϵ 1 , ϵ 2 Þ, 2Þ-Regular independent VDS is ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS, we have η rv ≤ ι rv : Since each minimum ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS is a minimal ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS, we have ι rv ≤ Y rv . The extensive activities of many countries in the world to produce electricity from wind energy have become an example for other countries. The economic exploitation of wind energy in electricity production is one of the new production methods in the world's electricity industry. The trend of wind power plant expansion shows a significant increase to reduce the cost of produced electricity. A wind turbine is a turbine that is used to convert the kinetic energy of the wind into mechanical or electrical energy, which is called wind power. It is made in two types: a horizontal axis and a vertical axis. Small wind turbines are used for applications such as charging batteries or auxiliary power in yachts, while larger wind turbines are used as a source of electrical energy by turning a generator and converting mechanical energy into electrical energy. In Iran, this capacity is also used to produce electricity. Wind power turbines have been installed and operated in the cities of Zabul, Mahshahr, Shiraz, Isfahan, Tabriz, Manjil, Binaloud, Khaf, Qazvin vineyards, and Ardebil.
The problem is how can we increase the amount of electricity produced with minimal wind turbines and have lower fuel costs. Which turbines are better to activate to reach the answer to the problem? To solve this problem, we first need to model the graph. The terms "amount of electricity produced" and "fuel cost reduction" are ambiguous in nature. Therefore, we need fuzzy graph modeling. Consider the vertices where the wind turbine is located and the edges denote the amount of energy production between them.
In Figure 7, the VG model shows the turbine installation locations and the amount of energy production between them. Consider T = fMahshahr, Zabul, Shiraz, Isfahan, Tabriz, Manjil, Binaloud, Khaf , Qazvin, Ardabilg as a set of    Tables 2 and 3.
In this VG, a DS S can be interpreted as a set of wind turbines that have more electricity production.
In this example, by activating at least wind turbines installed in the cities of Manjil, Ardebil, Zabul, and Khaf, the amount of electricity production can be increased, and the cost of fuel can be reduced.

4.2.
Application of a ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VWDS. In graph theory, the DS is an important issue in graphs. In this section, we explain the application of weak domination set in VG, and we present this concept in the form of an example. Suppose C is a VG (see Figure 8). In this example, we considered seven proposed points of a region for the construction of a clinic. From these seven suggested points, we are going to choose the minimum place for this work that meets the following conditions: good geographical location, facilities of the area, easy access to other places, and the possibility of development and expansion of the desired place. Suppose that X = fL 1 , L 2 , L 3 , L 4 , L 5 , L 6 , L 7 g are vertices and E = fL 1 L 2 , L 2 L 3 , L 3 L 4 , L 2 L 4 , L 4 L 5 , L 5 L 6 , L 4 L 6 , L 6 L 7 g are the edges of graph C.
In this VG, a DS S can be interpreted as a set of locations that have the best conditions. We have S = fL 1 , L 3 , L 5 , L 7 g that is a minimum size of ðð0:3,0:4Þ, 2Þ-Regular VWDS. Thus, η rvw = 2:05.
In this example, we can build clinics by choosing places that have the best conditions.

Conclusion
A VG is suitable for modeling problems with uncertainty which necessitates human knowledge and human evaluation. Moreover, dominations have a wide range of applications in VGs for the analysis of vague information and also serve as one of the most widely used topics in VGs in various sciences. In this research, we described a new concept of domination in VG called ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS. We also defined an independent strong (weak) DS in VG. Finally, an application of ððϵ 1 , ϵ 2 Þ, 2Þ-Regular VDS was presented. In future work, we will define a VG structure and study new types of domination, such as ððϵ 1 , ϵ 2 Þ, 2Þ-Regular DS and ððϵ 1 , ϵ 2 Þ, 2Þ-Regular independent DS on VG structure.

Data Availability
No data is used in this paper.

Conflicts of Interest
The authors declare that they have no conflicts of interest.  8 Advances in Mathematical Physics