Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 41, 2021, pp. 162-172 162 Perfect edge domination in vague graphs M. Kaliraja* P. Kanibose † A. Ibrahim‡ Abstract In this paper, we introduce the notions of perfect edge domination set and perfect edge domination number in vague graphs. Also, we introduce the definitions of connected perfect edge domination set and connected perfect edge dominating number. Moreover, we investigate some related properties of these concepts with comprehensive results and illustrations. Keywords: Vague graphs; Edge dominating set; Perfect edge domination set; Perfect edge domination number; Connected perfect edge domination set; Connected perfect edge domination number. 2010 AMS subject classification§: 03B60, 06B10, 06B20. * Assistant Professor, P.G. and Research Department of Mathematics, H. H. The Rajah’s College, Pudukkotai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India; mkr.maths009@gmail.com. †Research Scholar, P. G. and Research Department of Mathematics, H. H. The Rajah’s College, Pudukkotai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India; kanibose77@gmail.com. ‡ Assistant Professor, P.G. and Research Department of Mathematics, H. H. The Rajah’s College, Pudukkotai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India; ibrahimaadhil@yahoo.com; dribra@hhrc.ac.in § Received on June 25, 2021. Accepted on December 19, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.588. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Perfect edge domination in vague graphs 163 1. Introduction Edge domination sets in graphs are a phenomenon in research that has gained prominence due to the extreme scope of offering research solutions with varied dimensions. With the aid of perfect edge dominance and successful edge dominance in graphs, Chin Lung Lua, Ming-Tat Koa and Chuan Yi Tangb [3] investigated perfect edge dominance in graphs. R. S. Chitra and N. Prabhavathi [2] performed an analysis of perfect edges, perfect edge covering, and perfect edge vertex domination sets in graphs. The concept of ideal domination sets was proposed by S. Revathi, P.J. Jayalakshmi, and C.V. R. Harinarayanan [10].The notion of connected edge domination in fuzzy graphs was suggested by C.Y. Ponnappan, S. Basheer Ahamed, and P. Surulinathan [7]. P. Karpagam and V. Revathi [6] introduced the idea of connected edge perfect domination in fuzzy graphs with the idea of the principle of connected edge dominance in fuzzy graphs. S. Revathi, C.V.R. Harinarayanan, and R. Muthuraj [9] introduced an advanced idea of perfect domination in intuitionistic fuzzy graphs. W. L. Gau and D. J. Buehrer [4] proposed the concept of a vague set. R.A. Borzooeiy and H. Rashmanlou [1] introduced the notion of domination in vague graphs, and obtained strong domination numbers with applications. The definition of dominating sets in vague graphs was efficiently used in vague graphs by Yahya Talebi and Hossein Rashmanlou [11]. Recently, the authors [5] explored the concepts of edge dominance, independent edge domination in vague graphs, and obtained its related properties. In this paper, we present the notion of a perfect edge domination set and the perfect edge domination number of the vague graphs. Further, we introduce the connected perfect edge domination set and connected perfect edge dominating number. Also, we obtained some relevant properties. 2. Preliminaries In this section, we will show some basic definitions and properties that are helpful in developing our main results. Definition 2.1[4] A vague set 𝑃 in the universe of discourse 𝑋 is characterized by two membership functions given by i. A truth membership function 𝑡𝑃 : 𝑋 → [0, 1], ii. A false membership function 𝑓𝑃 : 𝑋 → [0, 1]. Where 𝑡𝑃 (𝑥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 𝑓𝑃 (𝑥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 𝑡𝑃 (𝑥) + 𝑓𝑃 (𝑥) ≤ 1. Thus the grade M. Kaliraja, P. Kanibose, A. Ibrahim 164 of membership of x in the vague set 𝑃 is bounded by a subinterval [ 𝑡𝑃 (𝑥), 1 − 𝑓𝑃 (𝑥)] of [0, 1]. The vague set 𝑃 is written as 𝑃 = {(𝑥, [ 𝑡𝑃 (𝑥), 𝑓𝑆 (𝑥)])/𝑥 ∈ 𝑋}, where the interval [ 𝑡𝑃 (𝑥), 1 − 𝑓𝑆 (𝑥)] is called the value of x in the vague set 𝑃. Definition 2.2[1] A vague graph is of the form 𝐺 = (𝑃, 𝑄), where i. A sequence of distinct vertices 𝑃 = {𝑣1, 𝑣2, … 𝑣𝑛 }, such that 𝑡𝑃 : 𝑃 → [0,1] and 𝑓𝑃 : 𝑃 → [0, 1] are truth and false membership functions, respectively such that 0≤ 𝑡𝑃 (𝑥) + 𝑓𝑃 (𝑥) ≤ 1. for all 𝑥 ∈ 𝑃. ii. A vague relation of the vague subsets 𝑋 × 𝑌 is an expression 𝑅, defined by 𝑅 = {{(𝑥, 𝑦), 𝑡𝑅 (𝑥, 𝑦), 𝑓𝑅 (𝑥, 𝑦)}/𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌}, where 𝑡𝑅 : 𝑋 × 𝑌 → [0,1] are𝑓𝑅 : 𝑋 × 𝑌 → [0,1], which satisfies the condition 0 ≤ 𝑡𝑅 (𝑥, 𝑦) + 𝑓𝑅 (𝑥, 𝑦) ≤ 1 for all (𝑥, 𝑦) ∈ 𝑋 × 𝑌. Definition 2.3[1] Let 𝐺 = (𝑃, 𝑄) be a vague graph, where 𝑃 = (𝑡𝑃,𝑓𝑃 ) is a vague set of vertex and 𝑄 = (𝑡𝑄,𝑓𝑄 ) is a set edge in 𝐺, then for every 𝑢𝑣 ∈ 𝑄, such that 𝑡𝑄 (𝑢𝑣) ≤ min{𝑡𝑄 (𝑢), 𝑡𝑄 (𝑣)} and 𝑓𝑄 (𝑢𝑣) ≤ max {𝑓𝑄 (𝑢), 𝑓𝑄 (𝑣)}. Definition 2.4 [1] Let 𝐺 = (𝑃, 𝑄) be a vague graph. If cardinality of the arcs 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) = min{𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} and 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) = max {𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} for all 𝑣𝑖 𝑣𝑗 ∈, then 𝐺 is called a complete vague graph. Definition 2.5[10] An arc of a vague graph 𝐺 = (𝑃, 𝑄) is said to be a strong edge in 𝐺. If 𝑡𝑄 (𝑢𝑣) ≥ (𝑡𝑄 ) ∞(𝑢𝑣) and 𝑓𝑄 (𝑢𝑣) ≤ (𝑓𝑄 ) ∞(𝑢𝑣). Where i. (𝑡𝑄 ) ∞(𝑢𝑣) = 𝑠𝑢𝑝{(𝑡𝑄 ) 𝑟 (𝑢𝑣): 𝑟 = 1,2, … , 𝑛} ii . (𝑓𝑄 ) ∞(𝑢𝑣) = 𝑖𝑛𝑓{(𝑓𝑄 ) 𝑟 (𝑢𝑣): 𝑟 = 1,2, … , 𝑛}. Definition 2.6[10] Let 𝑒𝑖 be an edge in a vague graph 𝐺 = (𝑃, 𝑄). Then the neighborhood of 𝑒𝑗 is representing by 𝑁(𝑒𝑖 ) = { 𝑒𝑗 ∈ 𝑄/(𝑢, 𝑣) is a strong arc} . Definition 2.7[7] Let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝑒𝑖 , 𝑒𝑗 ∈ 𝑄. If a strong arc 𝑒𝑖 is adjacent to 𝑒𝑗 . Then, we say that 𝑒𝑖, dominates 𝑒𝑗 . It is denoted by 𝐷. Definition 2.8[1] An edge 𝑒𝑗 in a vague graph 𝐺 = (𝑃, 𝑄) is called a neighbor of 𝑒𝑖 ∈ 𝐷 with respect to 𝐷, if 𝑁(𝑒𝑗 ) ∩ 𝐷 = {𝑒𝑖 }. Definition 2.9[1] Let 𝐺 = (𝑃, 𝑄) be a vague graph. If neighborhood degree is defined by i. The minimum neighborhood degree of 𝐺 is 𝛿(𝐺) = 𝑚𝑖𝑛{𝑑𝑁 (𝑒), 𝑒 ∈ 𝑄} ii. The maximum neighborhood degree of 𝐺 is ∆(𝐺) = 𝑚𝑎𝑥{𝑑𝑁 (𝑒), 𝑒 ∈ 𝑄}. Perfect edge domination in vague graphs 165 Definition 2.10[4] Two vertices 𝑣𝑖 and 𝑣𝑗 in a vague graph 𝐺 = (𝑃, 𝑄) are called a strong neighborhood 𝐺. If either one of the conditions are hold, i. 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) > 0, 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) > 0 ii. 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) = 0, 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) > 0 iii. 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) > 0, 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) = 0. Definition 2.11[10] Let 𝐺 = (𝑃, 𝑄) be a vague graph. Then number of edge (the cardinality of 𝑄) is called the order size of a vague graph and is denoted by 𝑂(𝑆) = ∑ ( 1+𝑡𝑄(𝑣𝑖𝑣𝑗)−𝑓𝑄(𝑣𝑖𝑣𝑗) 2 )𝑣𝑖𝑣𝑗∈𝑄 for all 𝑣𝑖 𝑣𝑗 ∈ 𝑄. Definition 2.12[4] Two edges in a vague graph 𝐺 = (𝑃, 𝑄) is celled an independent if there is no any strong arcs between them. Definition 2.13[4] Let 𝐺 = (𝑃, 𝑄) be a vague graph. If the sub graph is induced by 𝐷 has an isolated edge. Definition 2.14[4] A edge in vague graph 𝐺 = (𝑃, 𝑄)is an isolated edge, if it is not adjacent to any strong arc in 𝐺. 3. Main Results In this section, we introduce perfect edge domination set, perfect edge domination number, connected perfect edge domination set and connected perfect edge dominating number of vague graphs, and obtain some properties with illustrations. Definition 3.1 Let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝐷 be an edge dominating set in 𝐺. If for every edge of 𝑄(𝐺) − 𝐷 is adjacent to exactly one edge in 𝐷, then 𝐷 is called a perfect edge domination set in 𝐺. Example 3.2 Let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure 3.1. Consider the edge set 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,𝑒5}. We have 𝑒1,𝑒4 and 𝑒5 are strong arcs in 𝐺. M. Kaliraja, P. Kanibose, A. Ibrahim 166 Here, {𝑒5} and {𝑒1,𝑒4} are edge dominating sets in 𝐺. Now, 𝐷 = {𝑒5} is a perfect edge domination in 𝐺. Since, {𝑒5} is dominates the all other neighbor edges in 𝐺. Figure 3.1: Perfect edge domination set Definition 3.3 Let 𝐺 = (𝑃, 𝑄) be a vague graph. If it has a minimum perfect edge dominating set of vague graph 𝐺. Then it is called a perfect edge dominating number of 𝐺, and it is denoted by 𝛾𝑝(𝐷) in 𝐺. Example 3.4 Let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure 3.2. From the edge set 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,}, we see that {𝑒1} and {𝑒4} are strong arc 𝐺. Then {𝑒1} and {𝑒4} are a perfect edge dominating sets of 𝐺. Then minimum perfect edge dominating number is 𝛾𝑝(𝐷) = 0.70. Figure 3.2: Minimum perfect edge dominating number. Proposition 3.5 Let 𝐺 = (𝑃, 𝑄) be a vague graph with at least one isolated edge, then perfect edge dominating set does not exist. Proof: Let 𝐷 be a minimal perfect edge dominating set and 𝑒𝑖 be the path of 𝐷, since 𝐺 has at least one isolated edge. Incase 𝑄(𝐺) − 𝐷 is a perfect edge dominating set of vague graph, it has 𝑒𝑗 as neighborhood of perfect edge 𝑣3(0.2,0.6) 𝑣2(0.5,0.4) 𝑣1(0.3,0.4) 𝑒1(0.3,0.4) 𝑒3(0.2,0.8) 𝑣4 (0.5,0.3) 𝑒2(0.2,0.7) 𝑒4(0.3,0.5) 𝑒5(0.2,0.6) 𝑣1 (0.3,0.6) 𝑒4(0.2,0.5) (0.2,0.5) 𝑣4 (0.2,0.4) 𝑣3 (0.4,0.5) 𝑣2 (0.1,0.7) 𝑒1 (0.1,0.7) 𝑒3 (0.3,0.7) 𝑒2 (0.1,0.9) Perfect edge domination in vague graphs 167 domination set 𝐷. There exist a complement path of vague graph is denoted by 𝑁(𝑒𝑗 ). From the definition 2.6, we have |𝑁(𝑒𝑗 ) ∩ 𝐷|=1, but in this vague graph |𝑁(𝑒𝑗 ) ∩ 𝐷| ≠1. This is a contradiction. Therefore, 𝑄(𝐺) − 𝐷 not a perfect dominating set. ∎ Proposition 3.6 Let 𝐺 = (𝑃, 𝑄) be a complete vague graph. Then, every edge in 𝐷 is a perfect edge dominating set. Proof: Let 𝐺 = (𝑃, 𝑄)be a complete vague graph and let 𝑒𝑖 ∈ 𝐷 be a edge dominating set in 𝐺. Then every edges in the graph is 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) = min{𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} and 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) = max {𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} for all 𝑣𝑖 𝑣𝑗 ∈ 𝑄. Therefore, every path in 𝐺 is a strong arc and complement edges are 𝑒𝑗 ∈ 𝑄(𝐺) − 𝐷 adjacent to exactly one edge in 𝐷 is said to be a perfect edge domination set in vague graph. Here, 𝐷 is a perfect edge dominating set of vague graph 𝐺, and then every complete vague graph 𝐺 is a perfect edge dominating set. ∎ Example 3.7 Let 𝐺 = (𝑃, 𝑄) be a complete vague graph as shown in the figure 3.3. From the edge set 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,𝑒5} is an edge dominating set. Then, we have 𝑒1,𝑒4 and 𝑒5 which are strong arcs of the vague graph. Here {𝑒5} and {𝑒1,𝑒4} are edge dominating set of 𝐺, for 𝐷 = {𝑒5} is a perfect edge domination in vague graph. Figure 3.3: Perfect edge domination set 𝑣2(0.3, 0.6) 𝑣4(0.1, 0.6) 𝑣3 (0.4,0.5) 𝑣1 (0.2, 0.7) 𝑒1(0.1, 0.8) 𝑒2(0.4, 0.5) 𝑒3 (0.1, 0.6) 𝑒4(0.1, 0.7) 𝑒5(0.2, 0.7) 𝑒4 (0.1,0.7) M. Kaliraja, P. Kanibose, A. Ibrahim 168 Proposition 3.8 Let 𝐺 = (𝑃, 𝑄) be a vague graph, then 𝐷 is a minimal perfect edge dominating set. If for each 𝑄(𝐺) − 𝐷 is not a perfect edge dominating set. Proof: Let 𝐺 be a vague graph and has a minimal perfect edge domination set 𝐷. From the definition 3.1, if 𝐷 is minimum, the arcs must be strong. Suppose 𝑒𝑖 and 𝑒𝑗 are any two edges adjacent in 𝐺, but 𝑒𝑖 ∈ 𝐷 is a minimum perfect edge domination set of a vague graph. Then 𝑒𝑗 may or may not have any strong in this graph. Thus, each edge in 𝑒𝑗 has no strong neighbor of edge in 𝑄(𝐺) − 𝐷. Hence, 𝑄(𝐺) − 𝐷 is not perfect edge domination set of vague graph 𝐺. ∎ Definition 3.9 Let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝑆 is an edge dominating set of 𝐺 is connected perfect edge domination set with〈𝑆〉 is connected. Example 3.10 Let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure3.4. From the edge set to 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,𝑒5} , which are strong arcs of 𝐺. Here {𝑒1, 𝑒2} and {𝑒3,𝑒4} are edge dominating sets of 𝐺. Then 𝐷 = {𝑒1, 𝑒4} is connected perfect edge domination in 𝐺.Since, {𝑒1, 𝑒4} is a dominating set in 𝐺. Figure 3.4: Connected perfect edge domination Connected strong perfect edge dominating set 𝑄𝑐𝑠 = {𝑒1, 𝑒2} is connected, then 𝑄 − 𝑄𝑐𝑠 = {𝑒3, 𝑒4 , 𝑒5, 𝑒6} is connected 𝑁(𝑒3) = {𝑒2, 𝑒4 , 𝑒6 } ∩ {𝑒1, 𝑒2} = 𝑒2. 𝑁(𝑒5) = {𝑒1, 𝑒4 , 𝑒5 } ∩ {𝑒1, 𝑒2} = 𝑒1. 𝑣5(0.2,0.7) 𝑣1(0.3,0.6) 𝑣3(0.1,0.4) 𝑣2(0.1,0.8) 𝑒5(0.1,0.8) 𝑒6(0.1,0.8) 𝑒1(0.2,0.7) 𝑣4(0.4,0.5) 𝑒4(03,0.6) 𝑒3(0.1,0.5) 𝑒2(0.1,0.7) Perfect edge domination in vague graphs 169 𝑁(𝑒6) = {𝑒1, 𝑒5 , 𝑒3 } ∩ {𝑒1, 𝑒2} = 𝑒2. 𝑁(𝑒3) = {𝑒1, 𝑒5 , 𝑒3 } ∩ {𝑒1, 𝑒2} = 𝑒1. Definition 3.11 Let 𝐺 = (𝑃, 𝑄) be a vague graph. Then the smallest cardinality number of the arc in any edge connected perfect edge dominating set of 𝐺 is called connected perfect edge domination number, which is denoted as 𝛾𝑐𝑝. Proposition3.12 Let 𝐺 = (𝑃, 𝑄)be a vague graph and 〈𝑆〉 be a minimal connected edge perfect dominating set of 𝐺,then 𝑄(𝐺) − 〈𝑆〉 is also a connected edge perfect dominating set of 𝐺. Proof: Let {𝑒1, 𝑒2, 𝑒 3, … , 𝑒𝑛} be an edge set of vague graph 𝐺and 𝑀 be a minimal connected edge dominating set of 𝑄(𝐺). If 𝑄(𝐺) − 𝑀 is not a connected perfect edge dominating set. Then from, the definition 3.1, we have 𝐷 has minimum set of arcs that should be strong. if 𝑒𝑖 and 𝑒𝑗 are any two edges adjacent in 𝐺, but 𝑒𝑖 ∈ 𝐷 is a minimum perfect edge domination set of a vague graph, then 𝑒𝑗 is may (or) may not have any strong arc in this graph, Thus, every edge in 𝑒𝑗 has no strong neighbor of edge in 𝑄(𝐺) − 𝐷. Hence 𝑄(𝐺) − 𝐷 is not a perfect edge domination set of vague graph 𝐺. ∎ Example 3.13 Let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure 3.5. From the edge set 𝑄 = {𝑒1, 𝑒2, 𝑒3, 𝑒4, 𝑒5, 𝑒6,}, we see that 𝑒1,𝑒2 and 𝑒5 are strong arcs in the graph 𝐺.Then {𝑒1,𝑒5} are a connected perfect edge dominating set of 𝐺. Thus, {𝑒1,𝑒5} is minimum connected perfect edge dominating number is 𝛾𝑐𝑝(𝐷) = 0.50. . Figure 3.5: Minimum perfect edge dominating number 𝑣1(0.2,0.6) 𝑒1(0.1,0.6) 𝑣2(0.4,0.6) 𝑒2(0.1,0.7) 𝑣3(0.1,0.7) 𝑒4(0.1,0.8) 𝑣4(0.3,0.4) 𝑒6(0.2,0.6) 𝑣5(0.2,0.5) 𝑒5(0.2,0.7) 𝑒3(0.5,0.4) M. Kaliraja, P. Kanibose, A. Ibrahim 170 Remark 3.14 Let 𝐺 = (𝑃, 𝑄)be a vague graph and 〈𝑆〉 be a minimal connected edge perfect dominating set of 𝐺, then 𝑄(𝐺) − 〈𝑆〉 is a minimal connected edge perfect dominating set of 𝐺. Proposition 3.15 Let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝐷 is a connected perfect edge domination set and it is without isolated edges, then 𝑟 ∆𝑐(𝐺)+1 ≥ 𝛾𝐶𝑃 (𝐺). Proof: Let 𝐺 be a vague graph and 𝐷 be a connected perfect edge domination set. From the definition 3.9 of maximum neighbourhood and minimum neighbourhood of connected perfect edge domination set, then |𝐷|∆𝑐 (𝐺) ≤ ∑ 𝑑𝑄 (𝑒) = ∑|𝑁(𝑒)| 𝑒∈𝐷𝑒∈𝐷 ≤ |⋃ 𝑁(𝑒) 𝑒∈𝐷 | ≤ |𝑄 − 𝐷| ≤ 𝑟 − |𝐷| |𝐷|∆𝑐 (𝐺) + |𝐷| ≤ 𝑟 Therefore, 𝑟 ∆𝑐(𝐺)+1 ≥ 𝛾𝐶𝑃 (𝐺). ∎ Proposition 3.16 Every connected perfect edge dominating set of a vague graph 𝐺 = (𝑃, 𝑄) is not independent. Proof: Let 𝐺 be a vague graph and 𝐷 be a minimal connected perfect edge dominating set, then path of graphs have strong arcs. Clearly, every edges in the graph are satisfied by the condition of 𝑡𝑄 (𝑢𝑣) ≥ (𝑡𝑄 ) ∞(𝑢𝑣) and 𝑓𝑄 (𝑢𝑣) ≤ (𝑓𝑄 ) ∞(𝑢𝑣). Suppose, we assume that a vague graph 𝐷 contains an isolated edge 𝑒𝑖 in 𝐺. Since 𝐺 is a connected perfect edge dominating set, 𝑒𝑖 is a strong neighborhood of at least one edge 𝐷 − {𝑒𝑗 }. Thus, 𝐷 − {𝑒𝑗 } is an edge dominating set which contradicts in the minimum edge domination set 𝐷. Hence a vague graph is not independent. ∎ Proposition 3.17 Let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝑆𝑐 be a minimal connected perfect edge dominating set of 𝐺. Proof: Let 𝑆𝑐 be a minimal connected perfect edge dominating set of vague graph 𝐺 = (𝑃, 𝑄). If 𝑒𝑗 is not dominated by edge 𝑒𝑖 in 𝑆𝑐 and the induced vague graph < 𝑆𝑐 > is disconnected. We know that from the definition of Perfect edge domination in vague graphs 171 connected perfect dominating set, if every edge 𝑒𝑗 ∈ 𝑄 − 𝑆𝑐 is perfect dominated by exactly one edge 𝑒𝑖 in 𝑆𝑐 and the induced vague sub graph < 𝑆𝑐 > is connected. So, we have every edge 𝑒𝑗 ∈ 𝑄 − 𝑆𝑐 is dominated by some edge 𝑒𝑖 in 𝑆𝑐 which is a contradiction. Therefore, 𝑆𝑐 is connected dominating set. ∎ 4. Conclusions In this study, we have introduced the notions of perfect edge domination set and perfect edge domination number of vague graphs. Furthermore, we have investigated some related properties with suitable examples. 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