Edge Domination in Vague Graph

International Journal of Advanced Science and Technology Vol. 29, No. 1, (2020), pp. 1474 - 1480

M. Kaliraja1, P. Kanibose2 and A. Ibrahim3 1, 3P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University,Tamilnadu, India. 2Research Scholar, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University,Tamilnadu, India. [email protected] [email protected] [email protected]

Abstract In this paper, we introduce a notion ofastrong arc and edge domination set in a vague graph. Also, we determine edge domination number푑푒 (퐺) and independent dominating of a vague graph. Moreoverwe investigate somerelated properties in these concepts with illustrations.

Key words:Vague graph; Edge dominating set;Edge domination number;Edge independent set.

Mathematical Subject classification 2010:05C69, 05C99.

1.Introduction Graph theory is an important field of study in Mathematics with application to Communications, Computer science,Physical science, Bio-Science, and other areas. One the most important concept in this theory is dominations. This is widely applying real time incident. Dominating sets appear to have their origins in the game of chess. In 1965, L. A. Zadeh[13] first proposed the theory of fuzzy sets. But, Rosenfeld[10] introduced another elaborated definition including crisp relation, fuzzy sets, fuzzy vertex, fuzzy edge and several fuzzy digital of graph theoretic concepts. Ore[8]study of dominating set in graphs was stated. Zadeh[13] discuss the domination number and independent domination number.A. Somasundaram and S. Somasundaram[11] defined dominations in fuzzy graphs and define domination using effective edge in fuzzy graph.Atanssov[2] introduced the concept of intuitionistic fuzzy relations has been witnessing an growth in mathematical and its applications. S. Arumugam and S. Valammal[3] introduced the concept of connected edge domination of a connected graph. A. Nagoor Gani, J. Kavikumar and S. Anupriya[7]provide the edge domination on intuitionistic fuzzy graphs. W. L. Gau and D. J. Buehrer[8] proposed the notion of vague set in 1993, by replacing the value of an element in a set with a subinterval of [0,1].The study of vague graph by Ramakrishna [9] introduced the concept of vague graphs, and studied some of their properties. Rajab Ali Borzooeiy, Elham Darabianz, and Hossein Rashmanlou[4,5] obtained the vague graphs and strong domination numbers of vague graphs withapplications. Yahya Talebi and Hossein Rashmanlou[12]introduced the concept of application of dominating sets in vague graphs. In this paperpresent,we introduce definition of edge domination set using strong edge of vague graph. Further, we obtain edge domination number 푑푒 (퐺) and independent dominating of a vague graph. Moreover, we investigate some related properties of these concepts.

2. Preliminaries In this section, we recall some basic definitions and properties which are helpful to develop of main results.

Definition 2.1[13]Let퐴be a set. A function 휇: 푉 → [0, 1] is called a fuzzy subset on 퐴, for each 푥 ∈ 퐴, the value of 휇 푥 describes a degree of membership of 푥in 휇.

Definition 2.2[10]A fuzzy graph 퐺 = (휎, 휇) is a pair of functions 휎: 푉 → 0,1 and휇: 푉 × 푉 → 0,1 with 휇(푢, 푣) ≤ 휎(푢) ∧ 휎 푣 for all 푢, 푣 ∈ 푉, where 푉 is a finite non empty set and " ∧ " denote min { 휎(푢) ∧ 휎 푣 }.

Definition 2.3[11] Let 퐺 = (휎, 휇) be a fuzzy graph. A subset D of V is said to be a domination set of G if for every 푣 ∈ 푉 − 퐷 there exist 푢 ∈ 퐷 such that 휇 푢, 푣 = 휎(푢)˄휎(푣).

Definition 2.4[11] Let 퐺 = (휎, 휇) be a fuzzy graph. Let 푒푖 and 푒푗 be two edges of 퐺.We say that 푒푖 dominates 푒푗 is a strong arc in G and adjacent to 푒푗 . A subset 퐷 of 퐸(퐺) is said to be an edge dominating set of 퐺 if every 푒푗 ∈ 퐸 퐺 − 퐷. There exists 푒푖dominates 푒푗 .

Definition 2.5[9] A vague set 퐴 in the universe of discourse 푋 is characterized by two membership functions given by

(i) A truth membership function 푡퐴: 푋 → [0, 1],

(ii) A false membership function푓퐴: 푋 → [0, 1].

Where 푡퐴(푥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 푓퐴(푥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 푡퐴(푥)+푓퐴(푥) ≤ 1. Thus the grade of membership of x in the vague set 퐴 is bounded by a subinterval [푡퐴(푥), 1 − 푓퐴(푥)] of [0, 1]. The vague set 퐴 is written as 퐴 = {(푥, 푡퐴 푥 , 푓퐴 푥 )/푥 ∈ 푋}, where the interval [푡퐴(푥),

1 − 푓퐴(푥)] is called the value of x in the vague set 퐴 and denoted by 푉퐴 푥 .

Definition 2.6[9] A vague set 퐴 of a set 푋 is called

(i) the zero vague set of 푋 if 푡퐴 푥 = 0 and 푓퐴 푥 = 1 for all 푥 ∈ 푋,

(ii) the unit vague set of 푋 if 푡퐴 푥 = 1 and 푓퐴 푥 = 0 for all 푥 ∈ 푋.

Definition 2.7 [9]Let 퐺 = (푃, 푄) be a vague graph. Then the cardinality of G.vertex cardinality푃 and edge cardinality of 푄

1 + 푡 푣 − 푓 푣 1 + 푡푄 푣푖푣푗 − 푓푄(푣푖푣푗 ) G = 푃 푖 푃 푖 + 2 2 푣푖,∈푉 푣푖푣푗 ∈퐸 The vertex cardinality of G is defined by

1 + 푡 푣 − 푓 푣 푉 = 푃 푖 푃 푖 ; 푓표푟 푎푙푙 푣 ∈ 푉 2 푖 푣푖,∈푉

The edge cardinality of G is defined by

1 + 푡푄 푣푖푣푗 − 푓푄(푣푖푣푗 ) 퐸 = ; 푓표푟 푎푙푙(푣 , 푣 ) ∈ 퐸 2 푖 푗 푣푖푣푗 ∈퐸

The number of vertices is called the order of a vague graph and is denoted by 푂(퐺)and the number of edge is called the size of a vague graph and is denoted by 푂(푆).

Definition 2.8[5]Let 퐺 = 푃, 푄 be a vague graph, where 푃 = 푡푃, 푓푃 is a vague set on 푃 ⊆ 푄 × 푄 such that 푡푄 푥푦 ≤ min (푡푃 푥 , 푓푃 푦 and 푓푄 푥푦 ≥ max {푡푃 푥 , 푓푝 푦 } for all 푥푦 ∈ 퐸. A vague graph G is said to be strong if 푡푄 푣푖푣푗 = min 푡푃 푣푖 , fP(푣푗 ) and 푓푄 푣푖푣푗 = max⁡ 푡푃 vi , 푓푃 푣푗 for all푣푖푣푗 ∈ 푄.

Definition 2.9[4]Letu be a vertexin a vague graph퐺 = (푃, 푄).Then the neighborhood of 푢 is represent by 푁 푢 = 푣 ∈ 푉/(푢, 푣) is a strong arc .

Definition2.10[4]The strong neighborhood of an edge푒푖 in vague graph 퐺 = 푃, 푄 is푁푠 푒푖 = 푒푗 ∈ 퐸(퐺) 푒푗 is a strong arc in 퐺 and adjecent to 푒푖 .

Definition 2.11[4]An edge 푢, 푣 is said to be strong edge in vague graph 퐺 = 푃, 푄 if ∞ ∞ ∞ 푡푄 푢푣 ≥ 푡푄 푢푣 and 푓푄 푢푣 ≤ (푓푄) 푢푣 ,where 푡푄 푢푣 = 푘 ∞ 푘 max{ 푡푄 푢푣 : 푘 = 1,2, . . . , 푛, and (푓푄) (푢푣) = min {(푓푄) 푢푣 : 푘 = 1,2, . . . , 푛}.

∞ Definition 2.12[12]A subset S of V said to be independent set if 푡푄 푢푣 < 푡푄 (푢푣) and ∞ 푡푄 푢푣 > 푡푄 푢푣 .

Definition 2.13[12]Let 퐺 = 푃, 푄 be a vague graph, a subset 퐷 of 푄 said to be an edge dominations set in퐺.If for every in푄 − 퐷 is adjacent to D. The minimum vague graph cardinality of an edge dominating set in G is called the edge domination number of 퐺 and , is denoted by훾푒 (퐺).

Definition 2.14[12]Let퐺 = (푃, 푄)be a vague graph. For any 푢, 푣 ∈ 푃, we say that u dominate v in 퐺 if there exist a strong edge between them.

3. Edge Dominating Set of Vague Graph

In this section, we introduce edge domination of vague graphs and obtain some properties with illustrations.

Definition3.1Let 퐺 = (푃, 푄) be avague graphsand 푒푖, 푒푗 ∈ 푄.Then, we say that

푒푖dominates푒푗 , if 푒푖 is a strong arc in 퐺and adjacent to 푒푗 .

Example 3.2Let퐺 = (푃, 푄) be a vague graph, as shown in the figure 3.1. From the edge set 푄 = 푒1, 푒2, 푒3 , we have 푒1, 푒2 are strong arc of 푒3.

e3(0.3,0.5) Figure: 3.1 Definition 3.3Every edge in a vague graph 퐺 = (푃, 푄) is called an independent if there is no strong arc between them.

Example 3.4Let퐺 = (푃, 푄) be a vague graph, as shown in the figure 3.2. From the edge set 푄 = 푒1, 푒2, 푒3, 푒4 , we have 푒1, 푒2, 푒4, 푒4 there is no strong arc between them.

푒1(0.2,0.6) 푣4(0.2,0.3) 푣1(0.3,0.5)

푒3(0.1,0.5) 푒 (0.1,0.7) 2

푣3(0.4,0.5) 푣2(0.1,0.6) 푒 0.1,0.8 3 Figure: 3.2

Definition3.5Let 퐷 be a minimum dominating set of a vague graph 퐺.If 푒푗 ∈ 푄 퐺 − 퐷, then there exist 푒푖 ∈ 퐷 such that 푒푖 dominates 푒푗 edge.Thus, 퐷 is called an edge dominating set of 퐷.

Definition 3.6The minimum cardinality of all edge domination number of vague graph 퐺 and is denoted by 푑푒 (퐺)

Example 3.7 Let퐺 = (푃, 푄) be a vague graph, as shown in the figure 3.3, we have

(푒1, 푒5) is a minimal edge dominating set in cardinality. Therefore, we get푑푒 퐺 =0.60.

e1(02,0.6) v2(0.2,0.6) v1(0.3,0.4)

e6(0.3,0.6) e2(0.1,0.6)

v3(0.2,0.5)

e5(0.2,0.6) e3(0.1,0.6)

v5(0.4,0.5) v4(0.1,0.6) Figure:3.3 e4(0.1,0.6)

Note:For any 푒푖, 푒푗 ∈ 푄 퐺 . If 푒푖 dominate 푒푗 then 푒푗 dominate 푒푖.Thus, domination is a symmetric relation on 푄(퐺).

Proposition3.8Aedge dominating set 퐷 of a vague graph퐺 = (푃, 푄)is a edge minimal dominating set if and only if forevery푒푖 ∈ 퐷 one of the following conditions hold.

(i) 푒푖is not a strong neighbor of any edge in 퐷

(ii) There is a edge 푒푖 ∈ 푄 − 퐷 such that N(푒푖) ∩D = 푒 .

Proof.Let Dbe a minimal edge dominating set ofG.Then for every edge푒푖 ∈ 퐷,퐷 – 푒푖is not a dominatingset and hence there exists an edgee∈Q− (D − 푒푖 ) which isnot dominated by any edge in퐷 − 푒푖 . If 푆 = 푑, theneis not a strong neighbor of any edge in 퐷.If 퐷 − 푑 ,then eis not dominated by 퐷 − 푒푖 , but itis dominatedby 퐷. Then the edgeeis a strong neighbor only to din D. That is, 푁(푒) ∩ 퐷 = 푑. Conversely, assume that퐷is a edge dominating set and for each edge푑 ∈ 퐷, one ofthe two conditions holds. Suppose 퐷is not a minimaldominating set. Then there exists a edge 푑 ∈ 퐷, suchthat 퐷 − {푑}is a edge dominating set. Hence d is a strongneighbor to at least one edgein 퐷 − {푑}, and so (i)does not hold. If퐷 − {푑}is a edge dominating set, then everyedge in푄 − 퐷is a strong neighbor to at least oneedge in퐷 − {푑}, and so (ii) does not hold, which is acontradiction, since at least one of the conditions shouldbe hold. So,퐷is a minimal dominating set.∎

Proposition 3.9Let 퐷 be a minimal edge dominating set of connected vague graph 퐺 = (푃, 푄).Then,퐸(퐺) − 퐷 is an edge dominating set of 퐺 = 푃, 푄 .

Proof.Let 퐷 is a minimal edge set of vague graph 퐺 = (푃, 푄).Since 퐺 is connected. Then there exist 푒푖 ∈ 푄(퐺).It is dominated by at least one edge in푄(퐺) − 퐷 is dominating set. From the proposition 3.7,It follows that 푒푗 ∈ 푄 퐺 − 퐷.Thus every edge in 퐷 dominated by at least one edge in 푄 퐺 − 퐷 is dominating set. ∎

Proposition 3.10An edge independent set of an vague graph having only strong edge is a maximal edge independent set if and only if it is edge independent and edge dominating set.

Proof.Let퐷is a maximal edge independent set of 퐺 = (푃, 푄).Then푒푖 ∈ 푄 퐺 − 퐷.The set 퐷 ∪ 푒푖 is not independent for every 푒푖 ∈ 푄 퐺 − 퐷. There is anedge 푒푗 such that 푒푗 is strong neighbor to 푒푖 so 퐷 is a edge dominating set and also edge independent set of 퐺. Conversely, if퐷 is both edge independent and edge dominating set in G. Then,we have to prove 퐷 is maximal edge independent set having strong edges.Since,퐷is an edge dominating set having only strong edges, and if퐷 is not a maximal independent set. Then there exist edge푒푖not in 퐷 such that 퐷 ∪ 푒푖 is a dominating set, then no any edge in 퐷 is strong neighbor to 푒푖. Hence,퐷cannot be a dominating set, which is a contradiction. Thus,퐷 is maximal edge independent set of퐺 = 푃, 푄) having only strong edges. ∎

Proposition 3.11Every maximal edge independent set in a vague graph 퐺 = (푃, 푄)having only strong edges is a minimal edge dominating set of 퐺 = 푃, 푄 .

Proof.Let 퐷 be a maximal edge independent set having only strong edge of an vague graph 퐺(From the proposition 3.9) is a edge dominating set of 퐺.

Assume 퐷 is not minimal edge dominating set. Then there exist atleast one edge 푒푖in 퐷. such that 퐷 − 푒푖 dominates 푄 퐺 − 퐷 − 푒푖 is an 푁푠 푒푖 which is an contradiction. Therefore,퐷 is an independent set of 퐺. Hence, 퐷 is must be minimal dominating set. ∎

Proposition 3.12If 퐷 is an edge dominating set of a vague graph 퐺 which is containing at least one dominating set in G.

Proof.Let D be an edge dominating set of vague graph 퐺. Suppose edge dominating set 퐷 of a vague graph 퐺 no dominating set 퐷’in 퐺.From the proposition 3.8, if any two nodes of

퐷 are independent and non-adjacent. Then,푒푖 not in strong neighborhood of 푒푗 .Therefore, for every 푒푖 ∈ 푄 − 퐷.There exist no 푒푖 in 퐷,such that 푒푖 will not dominating 푒푗 . Which is a contraction to the edge domination set 퐷. Hence, we have 퐷 must contain at least one dominating set 퐷’in 퐺. ∎

Example 3.14 Let퐺 = (푃, 푄)be a vague graph, as shown in the figure 3.4.From 퐷 = 푒3,푒5 is an edge dominating set, Then 퐸 − 퐷 = 푒1, 푒2, 푒4, 푒6 ,Thus, 푒3,푒5 = 푣1, 푣3, 푣4, 푣5 .Let 퐷’= 푣1, 푣5 then 푉 − 퐷′ = 푣3, 푣5 Therefore 퐷′ is dominating set.

v1(0.2,0.4)

e2(0.2,0.5) e1(0.2,0.6)

v (0.3,0.6) 6 v2(o.3,0.4) (0.2,0.7) e (0.2,0.5) e6(0.1,0.6) 3 (0.1,0.5)

v 5(0.3,0.5) v (0.2,0.5) 3

e5(0.3,0.6) e4(0.1,0.6)

v4(0.4,0.5)

Figure:3.4

4 . Conclusion In this paper wehave introduced the notations of strong neighborhoodof and of vague graph. Further we introduced minimal dominating set in vague graph and edge dominating set. Also, we define independent set.Finally, we usingcardinality find the edge domination number of vague graph.

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