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A Study on Regular Domination in Vague Graphs with Application

A Study on Regular Domination in Vague Graphs with Application Hindawi Advances in Mathematical Physics Volume 2023, Article ID 7098134, 9 pages https://doi.org/10.1155/2023/7098134 Research Article A Study on Regular Domination in Vague Graphs with Application 1 2 3,4 3,4 Xiaolong Shi, Maryam Akhoundi , A. A. Talebi, and Masome Mojahedfar Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran Department of Mathematics, University of Mazandaran, Babolsar, Iran School of Mathematics, Damghan University, Damghan 3671641167, Iran Correspondence should be addressed to Maryam Akhoundi; maryam.akhoundi@mubabol.ac.ir Received 1 December 2022; Revised 9 March 2023; Accepted 2 May 2023; Published 20 May 2023 Academic Editor: S. A. Edalatpanah Copyright © 2023 Xiaolong Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In graph theory, a dominating set (DS) for a graph G = ðX, EÞ is a subset S of the vertices X such that every vertex not in S is adjacent to at least one member of S. The concept of DS in FGs has received the attention of many researchers due to its many applications in various fields such as computer science and electronic networks. In this paper, we introduce the notion of ððϵ , ϵ Þ,2Þ-Regular vague dominating set and provide some examples to explain various 1 2 concepts introduced. Also, some results were discussed. Additionally, the ððϵ , ϵ Þ,2Þ-Regular strong (weak) and independent 1 2 strong (weak) domination sets for vague domination set (VDS) were presented with some theorems to support the context. 1. Introduction 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 Zadeh [1] introduced the subject of a fuzzy set (FS) in 1995. Rosenfeld [2] proposed the subject of FGs. The definitions of Cockayne and Hedetniem [26]. In another study, A. Soma- FGs from the Zadeh fuzzy relations in 1973 were presented sundaram and S. Somasundaram [27] proposed the notion by Kaufmann [3]. Akram et al. [4–6] introduced several con- of domination in FGs. Kosari et al. [28] studied new con- cepts in FGs. Irregular VGs, domination in Pythagorean cepts in intuitionistic FG with an application in water sup- FGs, and 2-domination in VGs were studied by Banitalebi plier systems. et al. [7–9]. Gau and Buehrer [10] introduced the notion of Parvathi and Thamizhendhi [29] introduced the domi- a vague set (VS) in 1993. The concept of VGs was defined nation in intuitionistic FGs. Domination in product FGs by Ramakrishna [11]. Akram et al. [12] introduced vague and intuitionistic FGs was studied by Mahioub [30, 31]. Kar- hypergraphs. Rashmanlou et al. [13–15] investigated differ- unambigai et al. [32] introduced the domination in bipolar ent subjects of VGs. Moreover, Akram et al. [16–18] devel- FGs. Rao et al. [33–35] expressed certain properties of dom- oped several results on VGs. Kosari et al. [19] defined VG ination in vague incidence graphs. Shi and Kosari [36, 37] structure and studied its properties. The concepts of degree, studied the domination of product VGs with an application order, and size were developed by Gani and Begum [20]. in transportation. The concept of DS in FGs, both theoreti- Borzooei and Rashmanlou [21] proposed the degree of ver- cally and practically, is very valuable. A DS in FGs is used tices in VGs. Manjusha and Sunitha [22] studied the paired for solving problems of different branches in applied sci- domination. Haynes et al. [23] expressed the fundamentals ences such as location problems. In this way, the study of of domination in graphs. Nagoor Gani and Prasanna Devi new concepts such as DS is essential in FG. Domination in [24] suggested the reduction in the domination number of VGs has applications in several fields. Domination emerges 2 Advances in Mathematical Physics in the facility location problems, where the number of facil- Definition 5. Assume C = ðM, ZÞ is a VG on G , the degree ities is fixed and one endeavors to minimize the distance that of a vertex s is denoted as dðvÞ = ðd ðsÞ,d ðsÞÞ, where t f a person needs to travel to get to the closest facility. Qiang et al. [38] defined the novel concepts of domination in d ðÞ s = 〠 t ðÞ sv , t Z VGs. The notions of total domination, strong domination, s≠v,v∈X ð4Þ and connected domination in FGs using strong arcs were d s = 〠 f sv : ðÞ ðÞ studied by Manjusha and Sunitha [39–41]. Cockayne et al. f Z s≠v,v∈X [42] and Haynes et al. [43] investigated the independent and irredundance domination numbers in graphs. Natarajan The order of C is defined as and Ayyaswamy [44] introduced the notion of 2-strong (weak) domination in FGs. New results of irregular intuitio- nistic fuzzy graphs were presented by Talebi et al. [45, 46]. O C = 〠t s ,〠 f s : ð5Þ ðÞ ðÞ ðÞ M M Talebi and Rashmanlou, in [47], presented the concepts of s∈X s∈X DSs in VFGs. Narayanan and Murugesan [48] expressed the regular domination in intuitionistic fuzzy graph. A few ∗ Definition 6. Let C = ðM,ZÞ be a VG on G . A vertex s is researchers studied other domination variations which are called to dominate a vertex v if t ðsvÞ =min ft ðsÞ, t ðvÞg Z M M based on the above definitions such as independent domina- and f ðsvÞ =max ff ðsÞ, f ðvÞg: Z M M tion [49], complementary nil domination [50], and efficient domination [51]. In this paper, we introduced a new notion Definition 7. Let C = ðM, ZÞ be a VG on G . A subset S of of ððϵ , ϵ Þ,2Þ-Regular DS in VG. Finally, an application is 1 2 X is called to be VDS if there are some elements of S that given. dominate every vertex v ∈ X − S. 2. Preliminaries Definition 8. Let C = ðM, ZÞ be a VG on G . In this section, we present some preliminary results which (i) The vertex cardinality of C is defined by will be used throughout the paper. Definition 1. A graph G is a pair ðX, EÞ, where X is called 1+ t s − f s ðÞ ðÞ M M the vertex set and E ⊆ X × X is called the edge set. jj X =〠 : ð6Þ s∈X 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 (ii) The edge cardinality of C is defined by ζðÞ sv ≤ minfg ψðÞ s , ψðÞ v , ð1Þ 1+ t ðÞ sv − f ðÞ sv for all sv ∈ E: Z jj E = 〠 : ð7Þ sv∈E Definition 3 (see [10]). A vague set (VS) M is a pair ðt , f Þ on set X,where t and f are real-valued functions M M M Definition 9. Let C = ðM, ZÞ be a VG on G . which can be defined on X⟶ ½0, 1Š so that t ðsÞ + f ðsÞ ≤ M M The vertex cardinality of S ⊆ X of VG on G is defined by 1, ∀s ∈ X. 1+ t ðÞ s − f ðÞ s S = 〠 : ð8Þ jj Definition 4 (see [11]). A pair C = ðM, ZÞ is called a VG on s∈S graph G = ðX, EÞ, where M = ðt , f Þ is a VS on X and M M Z = ðt , f Þ is a VS on E such that Z Z Definition 10. Let C = ðM, ZÞ be a VG on G . The neighbor- hood of a vertex s ∈ X is defined by t sv ≤ min t s , t v , ðÞ fg ðÞ ðÞ Z M M ÈÉ ð2Þ N s = v ∈ X : t sv = t s ∧t s and f sv = f s ∨f s : ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ Z M M Z M M f sv ≥ max f s , f v , ðÞ fg ðÞ ðÞ Z M M ð9Þ for all s, v ∈ X. Note that Z is called vague relation on M.A The neighborhood degree (ND) is denoted by d ðsÞ and VG G is named strong if defined by t sv = min t s , t v , ðÞ fg ðÞ ðÞ Z M M 1+ t ðÞ sv − f ðÞ sv ð3Þ d s = 〠 : ð10Þ ðÞ f sv = max f s , f v , 2 ðÞ fg ðÞ ðÞ Z M M v∈N s ðÞ for all sv ∈ E: The minimum ND is δ ðCÞ = ∧fd ðsÞ: s ∈ Xg: N N Advances in Mathematical Physics 3 The maximum ND is Δ ðCÞ = ∨fd ðsÞ: s ∈ Xg: Table 1: Some essential notations. N N Notation Meaning Definition 11 (see [48]). Let C = ðM, ZÞ be a VG on G . Sup- FS Fuzzy set pose s and v are any two vertices in C. Then, v is called to FG Fuzzy graph strongly dominate s (s weakly dominate v)if VS Vague set (i) v dominate s VG Vague graph ND Neighborhood degree (ii) d ðvÞ ≥ d ðsÞ N N VDS Vague domination set VSDS Vague strong dominating set Definition 12 (see [48]). Let C = ðM, ZÞ be a VG on G . VWDS Vague weak dominating set X = fs ∈ X : d ðvÞ = δ ðCÞg and X = fs ∈ X : d ðvÞ = δ N N Δ N N N TDS Total dominating set Δ ðCÞg. VDN Vague domination number In Table 1, we show the essential notations. The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular 1 2 3. ððϵ , ϵ Þ,2Þ-Regular Domination in Vague VWDS is named ððϵ , ϵ Þ,2Þ-Regular vague weak domina- 1 2 1 2 tion number and is denoted by η ðCÞ. Graph rvw In this section, we define the notions of ððϵ , ϵ Þ,2Þ-Regular 1 2 Example 3. Consider a VG on G . DS, independent DS, and strong (weak) DS of VG. In Figure 3, S = fa, e, k, c, gg have a minimum size of VWDS, and each vertex in S has d = ð0:4,0:5Þ. Therefore, Definition 13. Let C = ðM, ZÞ be a VG on G . A subset S of S is ðð0:4,0:5Þ,2Þ-Regular VSDS. Thus, η =2:55. rvw X is called to be ððϵ , ϵ Þ,2Þ-Regular VDS if 1 2 Theorem 16. For a VG M, we have (i) every s ∈ X − S is dominated by two vertices in S (i) η ðCÞ ≤ η ðCÞ (ii) every vertex in S has degree ðϵ , ϵ Þ 1 2 rv rvs (ii) η ðCÞ ≤ η ðCÞ The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular rv rvw 1 2 VDS is named ððϵ , ϵ Þ,2Þ-Regular vague domination num- 1 2 ber (VDN) and denoted by η ðCÞ. rv Proof. Since every ððϵ , ϵ Þ,2Þ-Regular VSDS is a ððϵ , ϵ Þ, 1 2 1 2 2Þ-Regular VDS, we have η ðCÞ ≤ η ðCÞ. Further, since rv rvs Example 1. Consider a VG on G . every ððϵ , ϵ Þ,2Þ-Regular VWDS is a ððϵ , ϵ Þ,2Þ-Regular 1 2 1 2 VDS, we have η ðCÞ ≤ η ðCÞ. rv rvw In Figure 1, we have S = fb, e, g, kg and X − S = fa, c, d, f , hg. The vertices fe, b, kg dominate fa, c, dg, and also, Definition 17. Let C = ðM, ZÞ be a VG on G . A subset S of vertices fe, gg dominate ff , hg. We have d = ð0:4,0:8Þ. X is called to be ððϵ , ϵ Þ,2Þ-Regular independent VDS if Therefore, S is ðð0:4,0:8Þ,2Þ-Regular VDS. Thus, η =1:9. 1 2 rv (i) S is ððϵ , ϵ Þ,2Þ-Regular VDS 1 2 Definition 14. Let C = ðM, ZÞ be a VG on G . A set S ⊆ X is called to be ððϵ , ϵ Þ,2Þ-Regular vague strong dominating (ii) t ðsvÞ < t ðsÞ∧t ðsÞ and f ðsvÞ > f ðsÞ∨f ðsÞ, 1 2 Z M M Z M M set (VSDS) if every vertex s in V − S is strongly dominated for all s, v ∈ S. by two vertices of S and each vertex in S has degree ðϵ , ϵ Þ: The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular 1 2 1 2 The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular independent VDS is denoted by ι ðCÞ. 1 2 rv VSDS is named ððϵ , ϵ Þ,2Þ-Regular vague strong domina- 1 2 tion number and denoted by η ðCÞ. rvs Example 4. Consider a VG on G . In Figure 4, we have S = fy, tg that is ðð0:5,0:9Þ,2Þ-Reg- Example 2. Consider a VG on G . ular independent VDS. Therefore, ι =1. rv In Figure 2, we have S = fk, pg which is a minimum size of VSDS, and each vertex in S has d = ð0:8,0:6Þ. Therefore, Definition 18. Let C = ðM, ZÞ be a VG on G .A ððϵ , ϵ Þ,2Þ 1 2 S is ðð0:8,0:6Þ,2Þ-Regular VSDS. Thus, η =1:15. rvs -Regular VSDS is called to be ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 dent VSDS if S is independent. Definition 15. Let C = ðM, ZÞ be a VG on G . A set S ⊆ X is The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular 1 2 called to be ððϵ , ϵ Þ,2Þ-Regular vague weak dominating set independent VSDS is named ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 1 2 (VWDS) if every vertex s in V − S is weakly dominated by dent vague strong domination number and is denoted by two vertices of S and each vertex in S has degree ðϵ , ϵ Þ: ι ðCÞ. 1 2 rvs 4 Advances in Mathematical Physics (0.3, 0.4) (0.3, 0.4) (0.1, 0.4) (0.1, 0.2) (0.4, 0.2) (0.1, 0.2) (0.1, 0.2) (0.3, 0.4) (0.1, 0.2) (0.1, 0.4) (0.3, 0.4) (0.3, 0.4) (0.1, 0.2) e (0.2, 0.1) (0.1, 0.4) (0.3, 0.4) (0.1, 0.2) (0.1, 0.2) h (0.6, 0.2) Figure 1: ððϵ , ϵ Þ,2Þ-Regular VDS of VG. 1 2 (0.4, 0.1) (0.3, 0.1) We have S =1:85,S =1:8, S =1:65, S =1:65, (0.3, 0.1) 1 2 3 4 k S =1:8,and S =1:85. 5 6 Here, we see that S = fc, x, p, tg and S = fx, p, t, kg 3 4 have minimum size of ðð0:2,0:6Þ,2Þ-Regular independent (0.2, 0.1) (0.3, 0.4) VDS. Thus, ι =1:65. rvw (0.3, 0.1) Theorem 20. Let C = ðM,ZÞ be a VG on G .If S is ððϵ , (0.3, 0.1) ϵ Þ, 2Þ-Regular independent VWDS of C, then S ∩ X ≠∅. (0.3, 0.2) 2 δ s (0.2, 0.1) Proof. Assume s ∈ X . Since S is ððϵ , ϵ Þ,2Þ-Regular inde- δ 1 2 (0.2, 0.2) pendent VWDS, either s ∈ S or there exists a vertex v ∈ S (0.3, 0.2) such that t ðsvÞ = t ðsÞ∧t ðsÞ and f ðsvÞ = f ðsÞ∨f ðsÞ, Z M M Z M M for which d ðvÞ ≤ d ðsÞ.If s ∈ S,thenclearly S ∩ X ≠∅: N N δ (0.3, 0.2) On the other hand, ifd ðvÞ ≤ d ðsÞ,thend ðsÞ = δ ðCÞ. N N N N Therefore, S ∩ X ≠∅. Figure 2: ððϵ , ϵ Þ,2Þ-Regular VSDS of VG. N 1 2 Example 5. Consider a VG on G . Theorem 21. Let C = ðM,ZÞ be a VG on G .If S is ððϵ , In Figure 5, we have S = fp, s, k, qg of ðð0:3,0:6Þ,2Þ-Regu- ϵ Þ, 2Þ-Regular independent VSDS of C, then S ∩ X ≠∅. 2 Δ lar independent VDS which is a minimum size. Thus, ι =1:8. rvs Proof. Assume s ∈ X . Since S is ððϵ , ϵ Þ,2Þ-Regular inde- Δ 1 2 Definition 19. Let C = ðM, ZÞ be a VG on G .A ððϵ , ϵ Þ,2Þ 1 2 pendent VSDS, either s ∈ S or there exists a vertex v ∈ S -Regular VWDS is called to be ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 such that t ðsvÞ = t ðsÞ∧t ðsÞ and f ðsvÞ = f ðsÞ∨f ðsÞ, dent VWDS if S is independent. Z M M Z M M The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular for which d ðvÞ ≥ d ðsÞ.If s ∈ S,then clearly S ∩ X ≠∅: 1 2 N N Δ independent VWDS is named ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 On the other hand, if there exists a vertex v ∈ S,then v ∈ dent vague weak domination number and is denoted by X ðMÞ,because d ðsÞ = Δ ðCÞ.Therefore, S ∩ X ≠∅. Δ N N Δ N N ι ðCÞ. rvw Example 6. Consider a VG on G .In Figure6,wehavesix DS Theorem 22. Let C = ðM, ZÞ be a VG on G of order P. Then, ι ðCÞ ≤ P − Δ ðCÞ. rvs N S =fg k, w, c, x , Proof. Let S be ððϵ , ϵ Þ,2Þ-Regular independent VSDS. S =fg w, c, x, p , 2 1 2 Then, S ∩ X ≠∅. Suppose s ∈ S ∩ X . Since S is inde- Δ Δ N N S =fg c, x, p, t , pendent, S ∩ NðsÞ = ∅.So, S ⊆ X − NðsÞ, then, jSj ≤ jX − ð11Þ NðsÞj. Thus, ι ðCÞ ≤ P − Δ ðCÞ. S =fg x, p, t, k , 4 rvs N S =fg p, t, k, w , Theorem 23. Let C = ðM, ZÞ be a VG on G . Then, ι rvw S =fg t, k, w, c : ðCÞ ≤ P − δ ðCÞ. N Advances in Mathematical Physics 5 (0.2, 0.1) (0.2, 0.1) c g (0.2, 0.2) (0.2, 0.2) (0.2, 0.3) (0.2, 0.2) (0.2, 0.3) (0.4, 0.2) (0.2, 0.3) b d (0.4, 0.2) m n (0.4, 0.2) (0.3, 0.4) (0.2, 0.4) (0.2, 0.2) (0.2, 0.3) (0.4, 0.5) (0.4, 0.5) (0.4, 0.5) (0.2, 0.1) (0.4, 0.5) Figure 3: ððϵ , ϵ Þ,2Þ-Regular VWDS of VG. 1 2 mum size of minimal ððϵ , ϵ Þ,2Þ-Regular VDS is denoted (0.2, 0.3) 1 2 (0.2, 0.1) (0.2, 0.3) by Y ðCÞ: k rv (0.2, 0.3) Theorem 26. If C is ðϵ , ϵ Þ-Regular VG and S is minimal 1 2 (0.2, 0.3) (0.2, 0.3) ððϵ , ϵ Þ, 2Þ-Regular VDS, then X − S is ðϵ , ϵ Þ-Regular (0.2, 0.4) 1 2 1 2 VDS. Proof. Suppose S is a minimal ððϵ , ϵ Þ,2Þ-Regular VDS. 1 2 (0.2, 0.3) t (0.1, 0.2) (0.2, 0.1) Suppose X − S is not ðϵ , ϵ Þ-Regular VDS. Then, there is 1 2 v ∈ S that is not dominated by a vertex in X − S. Since C is a ðϵ , ϵ Þ-Regular VG, v must be dominated by two verti- 1 2 (0.1, 0.2) (0.1, 0.3) ces in S − fvg. Then, S − fvg is ððϵ , ϵ Þ,2Þ-Regular VDS 1 2 which is a contradiction. Hence, every vertex in S is domi- nated by two vertices in X − S. Therefore, X − S is ðϵ , ϵ Þ- 1 2 (0.1, 0.2) Regular VDS. Figure 4: ððϵ , ϵ Þ,2Þ-Regular independent VDS of VG. 1 2 Definition 27. Let C = ðM, ZÞ be a VG on G . A set S ⊆ X is named to be minimal ððϵ , ϵ Þ,2Þ-Regular VSDS (VWDS) if 1 2 Proof. Suppose S is ððϵ , ϵ Þ,2Þ-Regular independent 1 2 S − fsg is not a VSDS (VWDS). VWDS. Then, S ∩ X ≠∅. Suppose s ∈ S ∩ X . Since S δ δ N N is independent, S ∩ NðsÞ = ∅.So, S ⊆ X − NðsÞ,then, jSj ≤ Theorem 28. A ððϵ , ϵ Þ, 2Þ-Regular VDS C of a VG C is 1 2 jX − NðsÞj.Thus, ι ðCÞ ≤ P − δ ðCÞ. rvw N minimal if and only if for each s ∈ S, one of the following two conditions holds: Theorem 24. Let C = ðM, ZÞ be a complete VG on G with X = fs , s , ⋯, s g such that t ðs Þ = t ðs Þ≤⋯ ≤ t ðs Þ 1 2 n M 1 M 2 M n−1 (i) jNðsÞ ∩ Sj ≤ 1 ≤ t ðs Þ and f ðs Þ = f ðs Þ≥⋯≥ f ðs Þ ≥ f ðs Þ, and M n Z 1 Z 2 Z n−1 Z n (ii) There exists a vertex v ∈ X − S such that NðvÞ ∩ then, S = fs, tg, for a t ∈ S η C = 1 + t s − f s : ð12Þ ðÞ ðÞ ðÞ rvw M 1 M 1 Proof. Suppose S is a minimal ððϵ , ϵ Þ,2Þ-Regular domi- 1 2 Proof. Suppose C is a complete VG. Then, all edges are nating set. Then, for every vertex s ∈ S, S − fsg is not a strong. By hypothesis, d ðs Þ = d ðs Þ≤⋯≤ d ðs Þ = d N 1 N 2 N n−1 N ððϵ , ϵ Þ,2Þ-Regular VDS. This means that some vertex v ∈ 1 2 ðs Þ: Then, we get that fs , s g is a ððϵ , ϵ Þ,2Þ-Regular n 1 2 1 2 X − ðS − fsgÞ is not dominated by two vertices in S − fsg. VWDS with the minimum vague cardinality. Therefore, Then, either s = v or v ∈ X − S.If s = v,then jNðsÞ ∩ Sj ≤ 1. If s ≠ v,then v ∈ X − S.Since v is not dominated by S − fsg, η C = s , s =1+ t s − f s : ð13Þ ðÞ jj fg ðÞ ðÞ v is dominated by two vertices of s and t of S. Then, the vertex rvw 1 2 M 1 M 1 v is adjacent to s, t in S. Therefore, NðvÞ ∩ S = fs, tg.Con- versely, let S be ððϵ , ϵ Þ,2Þ-Regular VDS, and for every s ∈ 1 2 S, one of the two conditions holds. SupposeS is not a minimal Definition 25. A ððϵ , ϵ Þ,2Þ-Regular VDS of a graph C is dominating set. Then, there exists s ∈ S such that S − fsg is 1 2 called to be minimal ððϵ , ϵ Þ,2Þ-Regular VDS if it contains ððϵ , ϵ Þ,2Þ-Regular VDS. Hence, s is dominated by at least 1 2 1 2 no ððϵ , ϵ Þ,2Þ-Regular VDS as a proper subset. The maxi- two vertices in S − fsg. Therefore, condition (i) does not 1 2 6 Advances in Mathematical Physics (0.2, 0.3) (0.1, 0.2) (0.2, 0.1) (0.1, 0.3) (0.2, 0.3) (0.1, 0.3) (0.2, 0.3) (0.2, 0.3) (0.1, 0.2) (0.2, 0.3) (0.1, 0.2) (0.2, 0.3) (0.1, 0.3) (0.1, 0.2) (0.4, 0.3) (0.1, 0.2) (0.1, 0.2) Figure 5: ððϵ , ϵ Þ,2Þ-Regular independent VSDS of VG. 1 2 (0.1, 0.3) (0.1, 0.3) (0.1, 0.3) (0.2, 0.3) c (0.1, 0.3) (0.1, 0.3) (0.1, 0.3) (0.2, 0.1) (0.1, 0.3) (0.1, 0.3) (0.1, 0.3) (0.1, 0.2) Figure 6: ððϵ , ϵ Þ,2Þ-Regular independent VWDS of VG. 1 2 hold. Also, if S − fsg is ððϵ , ϵ Þ,2Þ-Regular VDS, then Definition 31. A ððϵ , ϵ Þ,2Þ-Regular independent VDS is 1 2 1 2 every vertex v in X − ðS − fsgÞ is dominated by at least called to be maximal ððϵ , ϵ Þ,2Þ-Regular independent 1 2 two vertices in S − fsg. Therefore, condition (ii) does not VDS, if every proper subset of S is not independent VDS. hold. This leads to a contradiction. Thus, S must be mini- mal ððϵ , ϵ Þ,2Þ-Regular VDS. Theorem 32. A ððϵ , ϵ Þ, 2Þ-Regular independent VDS is 1 2 1 2 maximal if and only if it is independent and ððϵ , ϵ Þ, 2Þ-Reg- 1 2 Similarly, we have the following theorem. ular VDS. Proof. Suppose S is maximal ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 Theorem 29. Suppose S is minimal ððϵ , ϵ Þ, 2Þ-Regular 1 2 dent VDS. It is trivial that S is ððϵ , ϵ Þ,2Þ-Regular VDS 1 2 VWDS. Then, for every s ∈ S, one of the following two condi- and S is independent. Conversely, consider that S is inde- tions holds: pendent and ððϵ , ϵ Þ,2Þ-Regular VDS. Suppose S is not 1 2 maximal; then, there exists v ∈ X − S such that S ∪ fvg is (i) There is no vertex other than s ∈ S independent. Then, v ∈ X − S is not adjacent to any vertex (ii) There exists vertex v ∈ X − S such that s and other in S which is a contradiction. Therefore, S is maximal. vertex t are the only vertices in S that weakly domi- nate v Theorem 33. In any VG C, η ≤ ι ≤ Y . rv rv rv Proof. Since each minimum ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 dent VDS is ððϵ , ϵ Þ,2Þ-Regular VDS, we have η ≤ ι : Definition 30. A ððϵ , ϵ Þ,2Þ-Regular VDS of a graph C is 1 2 1 2 rv rv called to be minimum ððϵ , ϵ Þ,2Þ-Regular VDS if it is DS Since each minimum ððϵ , ϵ Þ,2Þ-Regular VDS is a minimal 1 2 1 2 of minimum size. ððϵ , ϵ Þ,2Þ-Regular VDS, we have ι ≤ Y . 1 2 rv rv Advances in Mathematical Physics 7 (0.2, 0.1) (0.1, 0.2) Binaloud (0.2, 0.1) (0.1, 0.2) Mahshahr (0.2, 0.3) (0.1, 0.3) (0.2, 0.1) (0.2, 0.3) Ardabil Manjil (0.5, 0.3) Khaf (0.1, 0.3) (0.2, 0.3) (0.2, 0.3) (0.2, 0.3) (0.3, 0.3) (0.2, 0.3) (0.2, 0.3) (0.2, 0.1) Isfahan Zabul Tabriz (0.3, 0.3) (0.2, 0.3) (0.3, 0.3) (0.1, 0.3) (0.1, 0.3) Qazvin Shiraz (0.4, 0.3) (0.1, 0.3) Figure 7: VG of wind turbines. Table 2: The weights of vertices. 4. Application Zabul Mahshahr Shiraz Isfahan Tabriz 4.1. Application of a ððϵ , ϵ Þ, 2Þ-Regular Independent VDS. 1 2 0.3 0.1 0.1 0.2 0.3 The extensive activities of many countries in the world to produce electricity from wind energy have become an exam- 0.3 0.2 0.3 0.1 0.3 ple for other countries. The economic exploitation of wind Manjil Binaloud Khaf Qazvin Ardebil energy in electricity production is one of the new production 0.2 0.2 0.5 0.4 0.2 methods in the world’s electricity industry. The trend of 0.1 0.1 0.3 0.3 0.3 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 Table 3: The weights of edges. into mechanical or electrical energy, which is called wind power. It is made in two types: a horizontal axis and a verti- ðÞ t , f cal axis. 0:2, 0:3 Ardebil-Binaloud ðÞ Small wind turbines are used for applications such as 0:1, 0:2 Manjil-Mahshahr ðÞ charging batteries or auxiliary power in yachts, while larger wind turbines are used as a source of electrical energy by 0:3, 0:3 Khaf-Tabriz ðÞ turning a generator and converting mechanical energy into 0:1, 0:3 Shiraz-Zabul ðÞ electrical energy. In Iran, this capacity is also used to pro- ðÞ 0:2, 0:3 Qazvin-Isfahan duce electricity. Wind power turbines have been installed and operated in the cities of Zabul, Mahshahr, Shiraz, Isfa- ðÞ 0:2, 0:3 Khaf-Isfahan han, Tabriz, Manjil, Binaloud, Khaf, Qazvin vineyards, and ðÞ 0:2, 0:3 Zabul-Isfahan Ardebil. ðÞ 0:1, 0:3 Manjil-Shiraz The problem is how can we increase the amount of elec- tricity produced with minimal wind turbines and have lower ðÞ 0:2, 0:1 Binaloud-Manjil fuel costs. Which turbines are better to activate to reach the ðÞ 0:1, 0:3 Mahshahr-Khaf answer to the problem? To solve this problem, we first need ðÞ 0:1, 0:3 Tabriz-Shiraz to model the graph. The terms “amount of electricity pro- ðÞ 0:3, 0:3 duced” and “fuel cost reduction” are ambiguous in nature. Zabul-Qazvin Therefore, we need fuzzy graph modeling. Consider the ver- 0:2, 0:3 Isfahan-Ardebil ðÞ tices where the wind turbine is located and the edges denote 0:2, 0:3 Tabriz-Ardebil ðÞ the amount of energy production between them. 0:2, 0:3 Manjil-Qazvin ðÞ 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 8 Advances in Mathematical Physics (0.2, 0.1) L L 3 5 (0.1, 0.2) (0.2, 0.2) (0.2, 0.2) (0.1, 0.2) (0.3, 0.4) (0.3, 0.4) L L L L 1 4 L 2 6 (0.1, 0.2) (0.1, 0.2) (0.1, 0.2) (0.3, 0.4) (0.4, 0.2) (0.3, 0.2) (0.4, 0.4) Figure 8: VG of proposed locations. cities where the turbine is installed. The weights of the verti- Data Availability ces and edges are given in Tables 2 and 3. No data is used in this paper. In this VG, a DS S can be interpreted as a set of wind turbines that have more electricity production. We have S = fManjil, Ardebil, Zabul, Khaf g that is a Conflicts of Interest minimum size of ðð0:6,0:9Þ,2Þ-Regular independent VDS. Thus, ι =2:15. rvs The authors declare that they have no conflicts of interest. 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 Acknowledgments the cost of fuel can be reduced. This work was supported by the National Key R and D Pro- 4.2. Application of a ððϵ , ϵ Þ, 2Þ-Regular VWDS. In graph gram of China (Grant 2019YFA0706 402) and the National 1 2 theory, the DS is an important issue in graphs. In this sec- Natural Science Foundation of China under Grants tion, we explain the application of weak domination set in 62172302, 62072129, and 61876047. VG, and we present this concept in the form of an example. Suppose C is a VG (see Figure 8). In this example, we con- sidered seven proposed points of a region for the construc- References tion of a clinic. From these seven suggested points, we are [1] L. A. 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A Study on Regular Domination in Vague Graphs with Application

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10.1155/2023/7098134
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Hindawi Advances in Mathematical Physics Volume 2023, Article ID 7098134, 9 pages https://doi.org/10.1155/2023/7098134 Research Article A Study on Regular Domination in Vague Graphs with Application 1 2 3,4 3,4 Xiaolong Shi, Maryam Akhoundi , A. A. Talebi, and Masome Mojahedfar Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran Department of Mathematics, University of Mazandaran, Babolsar, Iran School of Mathematics, Damghan University, Damghan 3671641167, Iran Correspondence should be addressed to Maryam Akhoundi; maryam.akhoundi@mubabol.ac.ir Received 1 December 2022; Revised 9 March 2023; Accepted 2 May 2023; Published 20 May 2023 Academic Editor: S. A. Edalatpanah Copyright © 2023 Xiaolong Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In graph theory, a dominating set (DS) for a graph G = ðX, EÞ is a subset S of the vertices X such that every vertex not in S is adjacent to at least one member of S. The concept of DS in FGs has received the attention of many researchers due to its many applications in various fields such as computer science and electronic networks. In this paper, we introduce the notion of ððϵ , ϵ Þ,2Þ-Regular vague dominating set and provide some examples to explain various 1 2 concepts introduced. Also, some results were discussed. Additionally, the ððϵ , ϵ Þ,2Þ-Regular strong (weak) and independent 1 2 strong (weak) domination sets for vague domination set (VDS) were presented with some theorems to support the context. 1. Introduction 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 Zadeh [1] introduced the subject of a fuzzy set (FS) in 1995. Rosenfeld [2] proposed the subject of FGs. The definitions of Cockayne and Hedetniem [26]. In another study, A. Soma- FGs from the Zadeh fuzzy relations in 1973 were presented sundaram and S. Somasundaram [27] proposed the notion by Kaufmann [3]. Akram et al. [4–6] introduced several con- of domination in FGs. Kosari et al. [28] studied new con- cepts in FGs. Irregular VGs, domination in Pythagorean cepts in intuitionistic FG with an application in water sup- FGs, and 2-domination in VGs were studied by Banitalebi plier systems. et al. [7–9]. Gau and Buehrer [10] introduced the notion of Parvathi and Thamizhendhi [29] introduced the domi- a vague set (VS) in 1993. The concept of VGs was defined nation in intuitionistic FGs. Domination in product FGs by Ramakrishna [11]. Akram et al. [12] introduced vague and intuitionistic FGs was studied by Mahioub [30, 31]. Kar- hypergraphs. Rashmanlou et al. [13–15] investigated differ- unambigai et al. [32] introduced the domination in bipolar ent subjects of VGs. Moreover, Akram et al. [16–18] devel- FGs. Rao et al. [33–35] expressed certain properties of dom- oped several results on VGs. Kosari et al. [19] defined VG ination in vague incidence graphs. Shi and Kosari [36, 37] structure and studied its properties. The concepts of degree, studied the domination of product VGs with an application order, and size were developed by Gani and Begum [20]. in transportation. The concept of DS in FGs, both theoreti- Borzooei and Rashmanlou [21] proposed the degree of ver- cally and practically, is very valuable. A DS in FGs is used tices in VGs. Manjusha and Sunitha [22] studied the paired for solving problems of different branches in applied sci- domination. Haynes et al. [23] expressed the fundamentals ences such as location problems. In this way, the study of of domination in graphs. Nagoor Gani and Prasanna Devi new concepts such as DS is essential in FG. Domination in [24] suggested the reduction in the domination number of VGs has applications in several fields. Domination emerges 2 Advances in Mathematical Physics in the facility location problems, where the number of facil- Definition 5. Assume C = ðM, ZÞ is a VG on G , the degree ities is fixed and one endeavors to minimize the distance that of a vertex s is denoted as dðvÞ = ðd ðsÞ,d ðsÞÞ, where t f a person needs to travel to get to the closest facility. Qiang et al. [38] defined the novel concepts of domination in d ðÞ s = 〠 t ðÞ sv , t Z VGs. The notions of total domination, strong domination, s≠v,v∈X ð4Þ and connected domination in FGs using strong arcs were d s = 〠 f sv : ðÞ ðÞ studied by Manjusha and Sunitha [39–41]. Cockayne et al. f Z s≠v,v∈X [42] and Haynes et al. [43] investigated the independent and irredundance domination numbers in graphs. Natarajan The order of C is defined as and Ayyaswamy [44] introduced the notion of 2-strong (weak) domination in FGs. New results of irregular intuitio- nistic fuzzy graphs were presented by Talebi et al. [45, 46]. O C = 〠t s ,〠 f s : ð5Þ ðÞ ðÞ ðÞ M M Talebi and Rashmanlou, in [47], presented the concepts of s∈X s∈X DSs in VFGs. Narayanan and Murugesan [48] expressed the regular domination in intuitionistic fuzzy graph. A few ∗ Definition 6. Let C = ðM,ZÞ be a VG on G . A vertex s is researchers studied other domination variations which are called to dominate a vertex v if t ðsvÞ =min ft ðsÞ, t ðvÞg Z M M based on the above definitions such as independent domina- and f ðsvÞ =max ff ðsÞ, f ðvÞg: Z M M tion [49], complementary nil domination [50], and efficient domination [51]. In this paper, we introduced a new notion Definition 7. Let C = ðM, ZÞ be a VG on G . A subset S of of ððϵ , ϵ Þ,2Þ-Regular DS in VG. Finally, an application is 1 2 X is called to be VDS if there are some elements of S that given. dominate every vertex v ∈ X − S. 2. Preliminaries Definition 8. Let C = ðM, ZÞ be a VG on G . In this section, we present some preliminary results which (i) The vertex cardinality of C is defined by will be used throughout the paper. Definition 1. A graph G is a pair ðX, EÞ, where X is called 1+ t s − f s ðÞ ðÞ M M the vertex set and E ⊆ X × X is called the edge set. jj X =〠 : ð6Þ s∈X 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 (ii) The edge cardinality of C is defined by ζðÞ sv ≤ minfg ψðÞ s , ψðÞ v , ð1Þ 1+ t ðÞ sv − f ðÞ sv for all sv ∈ E: Z jj E = 〠 : ð7Þ sv∈E Definition 3 (see [10]). A vague set (VS) M is a pair ðt , f Þ on set X,where t and f are real-valued functions M M M Definition 9. Let C = ðM, ZÞ be a VG on G . which can be defined on X⟶ ½0, 1Š so that t ðsÞ + f ðsÞ ≤ M M The vertex cardinality of S ⊆ X of VG on G is defined by 1, ∀s ∈ X. 1+ t ðÞ s − f ðÞ s S = 〠 : ð8Þ jj Definition 4 (see [11]). A pair C = ðM, ZÞ is called a VG on s∈S graph G = ðX, EÞ, where M = ðt , f Þ is a VS on X and M M Z = ðt , f Þ is a VS on E such that Z Z Definition 10. Let C = ðM, ZÞ be a VG on G . The neighbor- hood of a vertex s ∈ X is defined by t sv ≤ min t s , t v , ðÞ fg ðÞ ðÞ Z M M ÈÉ ð2Þ N s = v ∈ X : t sv = t s ∧t s and f sv = f s ∨f s : ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ Z M M Z M M f sv ≥ max f s , f v , ðÞ fg ðÞ ðÞ Z M M ð9Þ for all s, v ∈ X. Note that Z is called vague relation on M.A The neighborhood degree (ND) is denoted by d ðsÞ and VG G is named strong if defined by t sv = min t s , t v , ðÞ fg ðÞ ðÞ Z M M 1+ t ðÞ sv − f ðÞ sv ð3Þ d s = 〠 : ð10Þ ðÞ f sv = max f s , f v , 2 ðÞ fg ðÞ ðÞ Z M M v∈N s ðÞ for all sv ∈ E: The minimum ND is δ ðCÞ = ∧fd ðsÞ: s ∈ Xg: N N Advances in Mathematical Physics 3 The maximum ND is Δ ðCÞ = ∨fd ðsÞ: s ∈ Xg: Table 1: Some essential notations. N N Notation Meaning Definition 11 (see [48]). Let C = ðM, ZÞ be a VG on G . Sup- FS Fuzzy set pose s and v are any two vertices in C. Then, v is called to FG Fuzzy graph strongly dominate s (s weakly dominate v)if VS Vague set (i) v dominate s VG Vague graph ND Neighborhood degree (ii) d ðvÞ ≥ d ðsÞ N N VDS Vague domination set VSDS Vague strong dominating set Definition 12 (see [48]). Let C = ðM, ZÞ be a VG on G . VWDS Vague weak dominating set X = fs ∈ X : d ðvÞ = δ ðCÞg and X = fs ∈ X : d ðvÞ = δ N N Δ N N N TDS Total dominating set Δ ðCÞg. VDN Vague domination number In Table 1, we show the essential notations. The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular 1 2 3. ððϵ , ϵ Þ,2Þ-Regular Domination in Vague VWDS is named ððϵ , ϵ Þ,2Þ-Regular vague weak domina- 1 2 1 2 tion number and is denoted by η ðCÞ. Graph rvw In this section, we define the notions of ððϵ , ϵ Þ,2Þ-Regular 1 2 Example 3. Consider a VG on G . DS, independent DS, and strong (weak) DS of VG. In Figure 3, S = fa, e, k, c, gg have a minimum size of VWDS, and each vertex in S has d = ð0:4,0:5Þ. Therefore, Definition 13. Let C = ðM, ZÞ be a VG on G . A subset S of S is ðð0:4,0:5Þ,2Þ-Regular VSDS. Thus, η =2:55. rvw X is called to be ððϵ , ϵ Þ,2Þ-Regular VDS if 1 2 Theorem 16. For a VG M, we have (i) every s ∈ X − S is dominated by two vertices in S (i) η ðCÞ ≤ η ðCÞ (ii) every vertex in S has degree ðϵ , ϵ Þ 1 2 rv rvs (ii) η ðCÞ ≤ η ðCÞ The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular rv rvw 1 2 VDS is named ððϵ , ϵ Þ,2Þ-Regular vague domination num- 1 2 ber (VDN) and denoted by η ðCÞ. rv Proof. Since every ððϵ , ϵ Þ,2Þ-Regular VSDS is a ððϵ , ϵ Þ, 1 2 1 2 2Þ-Regular VDS, we have η ðCÞ ≤ η ðCÞ. Further, since rv rvs Example 1. Consider a VG on G . every ððϵ , ϵ Þ,2Þ-Regular VWDS is a ððϵ , ϵ Þ,2Þ-Regular 1 2 1 2 VDS, we have η ðCÞ ≤ η ðCÞ. rv rvw In Figure 1, we have S = fb, e, g, kg and X − S = fa, c, d, f , hg. The vertices fe, b, kg dominate fa, c, dg, and also, Definition 17. Let C = ðM, ZÞ be a VG on G . A subset S of vertices fe, gg dominate ff , hg. We have d = ð0:4,0:8Þ. X is called to be ððϵ , ϵ Þ,2Þ-Regular independent VDS if Therefore, S is ðð0:4,0:8Þ,2Þ-Regular VDS. Thus, η =1:9. 1 2 rv (i) S is ððϵ , ϵ Þ,2Þ-Regular VDS 1 2 Definition 14. Let C = ðM, ZÞ be a VG on G . A set S ⊆ X is called to be ððϵ , ϵ Þ,2Þ-Regular vague strong dominating (ii) t ðsvÞ < t ðsÞ∧t ðsÞ and f ðsvÞ > f ðsÞ∨f ðsÞ, 1 2 Z M M Z M M set (VSDS) if every vertex s in V − S is strongly dominated for all s, v ∈ S. by two vertices of S and each vertex in S has degree ðϵ , ϵ Þ: The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular 1 2 1 2 The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular independent VDS is denoted by ι ðCÞ. 1 2 rv VSDS is named ððϵ , ϵ Þ,2Þ-Regular vague strong domina- 1 2 tion number and denoted by η ðCÞ. rvs Example 4. Consider a VG on G . In Figure 4, we have S = fy, tg that is ðð0:5,0:9Þ,2Þ-Reg- Example 2. Consider a VG on G . ular independent VDS. Therefore, ι =1. rv In Figure 2, we have S = fk, pg which is a minimum size of VSDS, and each vertex in S has d = ð0:8,0:6Þ. Therefore, Definition 18. Let C = ðM, ZÞ be a VG on G .A ððϵ , ϵ Þ,2Þ 1 2 S is ðð0:8,0:6Þ,2Þ-Regular VSDS. Thus, η =1:15. rvs -Regular VSDS is called to be ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 dent VSDS if S is independent. Definition 15. Let C = ðM, ZÞ be a VG on G . A set S ⊆ X is The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular 1 2 called to be ððϵ , ϵ Þ,2Þ-Regular vague weak dominating set independent VSDS is named ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 1 2 (VWDS) if every vertex s in V − S is weakly dominated by dent vague strong domination number and is denoted by two vertices of S and each vertex in S has degree ðϵ , ϵ Þ: ι ðCÞ. 1 2 rvs 4 Advances in Mathematical Physics (0.3, 0.4) (0.3, 0.4) (0.1, 0.4) (0.1, 0.2) (0.4, 0.2) (0.1, 0.2) (0.1, 0.2) (0.3, 0.4) (0.1, 0.2) (0.1, 0.4) (0.3, 0.4) (0.3, 0.4) (0.1, 0.2) e (0.2, 0.1) (0.1, 0.4) (0.3, 0.4) (0.1, 0.2) (0.1, 0.2) h (0.6, 0.2) Figure 1: ððϵ , ϵ Þ,2Þ-Regular VDS of VG. 1 2 (0.4, 0.1) (0.3, 0.1) We have S =1:85,S =1:8, S =1:65, S =1:65, (0.3, 0.1) 1 2 3 4 k S =1:8,and S =1:85. 5 6 Here, we see that S = fc, x, p, tg and S = fx, p, t, kg 3 4 have minimum size of ðð0:2,0:6Þ,2Þ-Regular independent (0.2, 0.1) (0.3, 0.4) VDS. Thus, ι =1:65. rvw (0.3, 0.1) Theorem 20. Let C = ðM,ZÞ be a VG on G .If S is ððϵ , (0.3, 0.1) ϵ Þ, 2Þ-Regular independent VWDS of C, then S ∩ X ≠∅. (0.3, 0.2) 2 δ s (0.2, 0.1) Proof. Assume s ∈ X . Since S is ððϵ , ϵ Þ,2Þ-Regular inde- δ 1 2 (0.2, 0.2) pendent VWDS, either s ∈ S or there exists a vertex v ∈ S (0.3, 0.2) such that t ðsvÞ = t ðsÞ∧t ðsÞ and f ðsvÞ = f ðsÞ∨f ðsÞ, Z M M Z M M for which d ðvÞ ≤ d ðsÞ.If s ∈ S,thenclearly S ∩ X ≠∅: N N δ (0.3, 0.2) On the other hand, ifd ðvÞ ≤ d ðsÞ,thend ðsÞ = δ ðCÞ. N N N N Therefore, S ∩ X ≠∅. Figure 2: ððϵ , ϵ Þ,2Þ-Regular VSDS of VG. N 1 2 Example 5. Consider a VG on G . Theorem 21. Let C = ðM,ZÞ be a VG on G .If S is ððϵ , In Figure 5, we have S = fp, s, k, qg of ðð0:3,0:6Þ,2Þ-Regu- ϵ Þ, 2Þ-Regular independent VSDS of C, then S ∩ X ≠∅. 2 Δ lar independent VDS which is a minimum size. Thus, ι =1:8. rvs Proof. Assume s ∈ X . Since S is ððϵ , ϵ Þ,2Þ-Regular inde- Δ 1 2 Definition 19. Let C = ðM, ZÞ be a VG on G .A ððϵ , ϵ Þ,2Þ 1 2 pendent VSDS, either s ∈ S or there exists a vertex v ∈ S -Regular VWDS is called to be ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 such that t ðsvÞ = t ðsÞ∧t ðsÞ and f ðsvÞ = f ðsÞ∨f ðsÞ, dent VWDS if S is independent. Z M M Z M M The minimum vague cardinality of ððϵ , ϵ Þ,2Þ-Regular for which d ðvÞ ≥ d ðsÞ.If s ∈ S,then clearly S ∩ X ≠∅: 1 2 N N Δ independent VWDS is named ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 On the other hand, if there exists a vertex v ∈ S,then v ∈ dent vague weak domination number and is denoted by X ðMÞ,because d ðsÞ = Δ ðCÞ.Therefore, S ∩ X ≠∅. Δ N N Δ N N ι ðCÞ. rvw Example 6. Consider a VG on G .In Figure6,wehavesix DS Theorem 22. Let C = ðM, ZÞ be a VG on G of order P. Then, ι ðCÞ ≤ P − Δ ðCÞ. rvs N S =fg k, w, c, x , Proof. Let S be ððϵ , ϵ Þ,2Þ-Regular independent VSDS. S =fg w, c, x, p , 2 1 2 Then, S ∩ X ≠∅. Suppose s ∈ S ∩ X . Since S is inde- Δ Δ N N S =fg c, x, p, t , pendent, S ∩ NðsÞ = ∅.So, S ⊆ X − NðsÞ, then, jSj ≤ jX − ð11Þ NðsÞj. Thus, ι ðCÞ ≤ P − Δ ðCÞ. S =fg x, p, t, k , 4 rvs N S =fg p, t, k, w , Theorem 23. Let C = ðM, ZÞ be a VG on G . Then, ι rvw S =fg t, k, w, c : ðCÞ ≤ P − δ ðCÞ. N Advances in Mathematical Physics 5 (0.2, 0.1) (0.2, 0.1) c g (0.2, 0.2) (0.2, 0.2) (0.2, 0.3) (0.2, 0.2) (0.2, 0.3) (0.4, 0.2) (0.2, 0.3) b d (0.4, 0.2) m n (0.4, 0.2) (0.3, 0.4) (0.2, 0.4) (0.2, 0.2) (0.2, 0.3) (0.4, 0.5) (0.4, 0.5) (0.4, 0.5) (0.2, 0.1) (0.4, 0.5) Figure 3: ððϵ , ϵ Þ,2Þ-Regular VWDS of VG. 1 2 mum size of minimal ððϵ , ϵ Þ,2Þ-Regular VDS is denoted (0.2, 0.3) 1 2 (0.2, 0.1) (0.2, 0.3) by Y ðCÞ: k rv (0.2, 0.3) Theorem 26. If C is ðϵ , ϵ Þ-Regular VG and S is minimal 1 2 (0.2, 0.3) (0.2, 0.3) ððϵ , ϵ Þ, 2Þ-Regular VDS, then X − S is ðϵ , ϵ Þ-Regular (0.2, 0.4) 1 2 1 2 VDS. Proof. Suppose S is a minimal ððϵ , ϵ Þ,2Þ-Regular VDS. 1 2 (0.2, 0.3) t (0.1, 0.2) (0.2, 0.1) Suppose X − S is not ðϵ , ϵ Þ-Regular VDS. Then, there is 1 2 v ∈ S that is not dominated by a vertex in X − S. Since C is a ðϵ , ϵ Þ-Regular VG, v must be dominated by two verti- 1 2 (0.1, 0.2) (0.1, 0.3) ces in S − fvg. Then, S − fvg is ððϵ , ϵ Þ,2Þ-Regular VDS 1 2 which is a contradiction. Hence, every vertex in S is domi- nated by two vertices in X − S. Therefore, X − S is ðϵ , ϵ Þ- 1 2 (0.1, 0.2) Regular VDS. Figure 4: ððϵ , ϵ Þ,2Þ-Regular independent VDS of VG. 1 2 Definition 27. Let C = ðM, ZÞ be a VG on G . A set S ⊆ X is named to be minimal ððϵ , ϵ Þ,2Þ-Regular VSDS (VWDS) if 1 2 Proof. Suppose S is ððϵ , ϵ Þ,2Þ-Regular independent 1 2 S − fsg is not a VSDS (VWDS). VWDS. Then, S ∩ X ≠∅. Suppose s ∈ S ∩ X . Since S δ δ N N is independent, S ∩ NðsÞ = ∅.So, S ⊆ X − NðsÞ,then, jSj ≤ Theorem 28. A ððϵ , ϵ Þ, 2Þ-Regular VDS C of a VG C is 1 2 jX − NðsÞj.Thus, ι ðCÞ ≤ P − δ ðCÞ. rvw N minimal if and only if for each s ∈ S, one of the following two conditions holds: Theorem 24. Let C = ðM, ZÞ be a complete VG on G with X = fs , s , ⋯, s g such that t ðs Þ = t ðs Þ≤⋯ ≤ t ðs Þ 1 2 n M 1 M 2 M n−1 (i) jNðsÞ ∩ Sj ≤ 1 ≤ t ðs Þ and f ðs Þ = f ðs Þ≥⋯≥ f ðs Þ ≥ f ðs Þ, and M n Z 1 Z 2 Z n−1 Z n (ii) There exists a vertex v ∈ X − S such that NðvÞ ∩ then, S = fs, tg, for a t ∈ S η C = 1 + t s − f s : ð12Þ ðÞ ðÞ ðÞ rvw M 1 M 1 Proof. Suppose S is a minimal ððϵ , ϵ Þ,2Þ-Regular domi- 1 2 Proof. Suppose C is a complete VG. Then, all edges are nating set. Then, for every vertex s ∈ S, S − fsg is not a strong. By hypothesis, d ðs Þ = d ðs Þ≤⋯≤ d ðs Þ = d N 1 N 2 N n−1 N ððϵ , ϵ Þ,2Þ-Regular VDS. This means that some vertex v ∈ 1 2 ðs Þ: Then, we get that fs , s g is a ððϵ , ϵ Þ,2Þ-Regular n 1 2 1 2 X − ðS − fsgÞ is not dominated by two vertices in S − fsg. VWDS with the minimum vague cardinality. Therefore, Then, either s = v or v ∈ X − S.If s = v,then jNðsÞ ∩ Sj ≤ 1. If s ≠ v,then v ∈ X − S.Since v is not dominated by S − fsg, η C = s , s =1+ t s − f s : ð13Þ ðÞ jj fg ðÞ ðÞ v is dominated by two vertices of s and t of S. Then, the vertex rvw 1 2 M 1 M 1 v is adjacent to s, t in S. Therefore, NðvÞ ∩ S = fs, tg.Con- versely, let S be ððϵ , ϵ Þ,2Þ-Regular VDS, and for every s ∈ 1 2 S, one of the two conditions holds. SupposeS is not a minimal Definition 25. A ððϵ , ϵ Þ,2Þ-Regular VDS of a graph C is dominating set. Then, there exists s ∈ S such that S − fsg is 1 2 called to be minimal ððϵ , ϵ Þ,2Þ-Regular VDS if it contains ððϵ , ϵ Þ,2Þ-Regular VDS. Hence, s is dominated by at least 1 2 1 2 no ððϵ , ϵ Þ,2Þ-Regular VDS as a proper subset. The maxi- two vertices in S − fsg. Therefore, condition (i) does not 1 2 6 Advances in Mathematical Physics (0.2, 0.3) (0.1, 0.2) (0.2, 0.1) (0.1, 0.3) (0.2, 0.3) (0.1, 0.3) (0.2, 0.3) (0.2, 0.3) (0.1, 0.2) (0.2, 0.3) (0.1, 0.2) (0.2, 0.3) (0.1, 0.3) (0.1, 0.2) (0.4, 0.3) (0.1, 0.2) (0.1, 0.2) Figure 5: ððϵ , ϵ Þ,2Þ-Regular independent VSDS of VG. 1 2 (0.1, 0.3) (0.1, 0.3) (0.1, 0.3) (0.2, 0.3) c (0.1, 0.3) (0.1, 0.3) (0.1, 0.3) (0.2, 0.1) (0.1, 0.3) (0.1, 0.3) (0.1, 0.3) (0.1, 0.2) Figure 6: ððϵ , ϵ Þ,2Þ-Regular independent VWDS of VG. 1 2 hold. Also, if S − fsg is ððϵ , ϵ Þ,2Þ-Regular VDS, then Definition 31. A ððϵ , ϵ Þ,2Þ-Regular independent VDS is 1 2 1 2 every vertex v in X − ðS − fsgÞ is dominated by at least called to be maximal ððϵ , ϵ Þ,2Þ-Regular independent 1 2 two vertices in S − fsg. Therefore, condition (ii) does not VDS, if every proper subset of S is not independent VDS. hold. This leads to a contradiction. Thus, S must be mini- mal ððϵ , ϵ Þ,2Þ-Regular VDS. Theorem 32. A ððϵ , ϵ Þ, 2Þ-Regular independent VDS is 1 2 1 2 maximal if and only if it is independent and ððϵ , ϵ Þ, 2Þ-Reg- 1 2 Similarly, we have the following theorem. ular VDS. Proof. Suppose S is maximal ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 Theorem 29. Suppose S is minimal ððϵ , ϵ Þ, 2Þ-Regular 1 2 dent VDS. It is trivial that S is ððϵ , ϵ Þ,2Þ-Regular VDS 1 2 VWDS. Then, for every s ∈ S, one of the following two condi- and S is independent. Conversely, consider that S is inde- tions holds: pendent and ððϵ , ϵ Þ,2Þ-Regular VDS. Suppose S is not 1 2 maximal; then, there exists v ∈ X − S such that S ∪ fvg is (i) There is no vertex other than s ∈ S independent. Then, v ∈ X − S is not adjacent to any vertex (ii) There exists vertex v ∈ X − S such that s and other in S which is a contradiction. Therefore, S is maximal. vertex t are the only vertices in S that weakly domi- nate v Theorem 33. In any VG C, η ≤ ι ≤ Y . rv rv rv Proof. Since each minimum ððϵ , ϵ Þ,2Þ-Regular indepen- 1 2 dent VDS is ððϵ , ϵ Þ,2Þ-Regular VDS, we have η ≤ ι : Definition 30. A ððϵ , ϵ Þ,2Þ-Regular VDS of a graph C is 1 2 1 2 rv rv called to be minimum ððϵ , ϵ Þ,2Þ-Regular VDS if it is DS Since each minimum ððϵ , ϵ Þ,2Þ-Regular VDS is a minimal 1 2 1 2 of minimum size. ððϵ , ϵ Þ,2Þ-Regular VDS, we have ι ≤ Y . 1 2 rv rv Advances in Mathematical Physics 7 (0.2, 0.1) (0.1, 0.2) Binaloud (0.2, 0.1) (0.1, 0.2) Mahshahr (0.2, 0.3) (0.1, 0.3) (0.2, 0.1) (0.2, 0.3) Ardabil Manjil (0.5, 0.3) Khaf (0.1, 0.3) (0.2, 0.3) (0.2, 0.3) (0.2, 0.3) (0.3, 0.3) (0.2, 0.3) (0.2, 0.3) (0.2, 0.1) Isfahan Zabul Tabriz (0.3, 0.3) (0.2, 0.3) (0.3, 0.3) (0.1, 0.3) (0.1, 0.3) Qazvin Shiraz (0.4, 0.3) (0.1, 0.3) Figure 7: VG of wind turbines. Table 2: The weights of vertices. 4. Application Zabul Mahshahr Shiraz Isfahan Tabriz 4.1. Application of a ððϵ , ϵ Þ, 2Þ-Regular Independent VDS. 1 2 0.3 0.1 0.1 0.2 0.3 The extensive activities of many countries in the world to produce electricity from wind energy have become an exam- 0.3 0.2 0.3 0.1 0.3 ple for other countries. The economic exploitation of wind Manjil Binaloud Khaf Qazvin Ardebil energy in electricity production is one of the new production 0.2 0.2 0.5 0.4 0.2 methods in the world’s electricity industry. The trend of 0.1 0.1 0.3 0.3 0.3 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 Table 3: The weights of edges. into mechanical or electrical energy, which is called wind power. It is made in two types: a horizontal axis and a verti- ðÞ t , f cal axis. 0:2, 0:3 Ardebil-Binaloud ðÞ Small wind turbines are used for applications such as 0:1, 0:2 Manjil-Mahshahr ðÞ charging batteries or auxiliary power in yachts, while larger wind turbines are used as a source of electrical energy by 0:3, 0:3 Khaf-Tabriz ðÞ turning a generator and converting mechanical energy into 0:1, 0:3 Shiraz-Zabul ðÞ electrical energy. In Iran, this capacity is also used to pro- ðÞ 0:2, 0:3 Qazvin-Isfahan duce electricity. Wind power turbines have been installed and operated in the cities of Zabul, Mahshahr, Shiraz, Isfa- ðÞ 0:2, 0:3 Khaf-Isfahan han, Tabriz, Manjil, Binaloud, Khaf, Qazvin vineyards, and ðÞ 0:2, 0:3 Zabul-Isfahan Ardebil. ðÞ 0:1, 0:3 Manjil-Shiraz The problem is how can we increase the amount of elec- tricity produced with minimal wind turbines and have lower ðÞ 0:2, 0:1 Binaloud-Manjil fuel costs. Which turbines are better to activate to reach the ðÞ 0:1, 0:3 Mahshahr-Khaf answer to the problem? To solve this problem, we first need ðÞ 0:1, 0:3 Tabriz-Shiraz to model the graph. The terms “amount of electricity pro- ðÞ 0:3, 0:3 duced” and “fuel cost reduction” are ambiguous in nature. Zabul-Qazvin Therefore, we need fuzzy graph modeling. Consider the ver- 0:2, 0:3 Isfahan-Ardebil ðÞ tices where the wind turbine is located and the edges denote 0:2, 0:3 Tabriz-Ardebil ðÞ the amount of energy production between them. 0:2, 0:3 Manjil-Qazvin ðÞ 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 8 Advances in Mathematical Physics (0.2, 0.1) L L 3 5 (0.1, 0.2) (0.2, 0.2) (0.2, 0.2) (0.1, 0.2) (0.3, 0.4) (0.3, 0.4) L L L L 1 4 L 2 6 (0.1, 0.2) (0.1, 0.2) (0.1, 0.2) (0.3, 0.4) (0.4, 0.2) (0.3, 0.2) (0.4, 0.4) Figure 8: VG of proposed locations. cities where the turbine is installed. The weights of the verti- Data Availability ces and edges are given in Tables 2 and 3. No data is used in this paper. In this VG, a DS S can be interpreted as a set of wind turbines that have more electricity production. We have S = fManjil, Ardebil, Zabul, Khaf g that is a Conflicts of Interest minimum size of ðð0:6,0:9Þ,2Þ-Regular independent VDS. Thus, ι =2:15. rvs The authors declare that they have no conflicts of interest. 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 Acknowledgments the cost of fuel can be reduced. This work was supported by the National Key R and D Pro- 4.2. Application of a ððϵ , ϵ Þ, 2Þ-Regular VWDS. In graph gram of China (Grant 2019YFA0706 402) and the National 1 2 theory, the DS is an important issue in graphs. In this sec- Natural Science Foundation of China under Grants tion, we explain the application of weak domination set in 62172302, 62072129, and 61876047. VG, and we present this concept in the form of an example. Suppose C is a VG (see Figure 8). In this example, we con- sidered seven proposed points of a region for the construc- References tion of a clinic. From these seven suggested points, we are [1] L. A. 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