Edge Domination in Vague Graph
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.
Abstract
In this paper, we introduce a notion ofastrong arc and edge domination set in a vague graph. Also, we determine edge domination number𝑑𝑒(𝐺) 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 𝑑𝑒(𝐺) 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]Let𝐴be a set. A function 𝜇: 𝑉 → [0, 1] is called a fuzzy subset on 𝐴, for each 𝑥 ∈ 𝐴, the value of 𝜇 𝑥 describes a degree of membership of 𝑥in 𝜇.
Definition 2.2[10]A fuzzy graph 𝐺 = (𝜎, 𝜇) is a pair of functions 𝜎: 𝑉 → 0,1 and𝜇: 𝑉 × 𝑉 → 0,1 with 𝜇(𝑢, 𝑣) ≤ 𝜎(𝑢) ∧ 𝜎 𝑣 for all 𝑢, 𝑣 ∈ 𝑉, where 𝑉 is a finite non empty set and " ∧ " denote min { 𝜎(𝑢) ∧ 𝜎 𝑣 }.
Definition 2.3[11] Let 𝐺 = (𝜎, 𝜇) be a fuzzy graph. A subset D of V is said to be a domination set of G if for every 𝑣 ∈ 𝑉 − 𝐷 there exist 𝑢 ∈ 𝐷 such that 𝜇 𝑢, 𝑣 = 𝜎(𝑢)˄𝜎(𝑣).
Definition 2.4[11] Let 𝐺 = (𝜎, 𝜇) be a fuzzy graph. Let 𝑒𝑖 and 𝑒𝑗 be two edges of 𝐺.We say that 𝑒𝑖 dominates 𝑒𝑗 is a strong arc in G and adjacent to 𝑒𝑗. A subset 𝐷 of 𝐸(𝐺) is said to be an edge dominating set of 𝐺 if every 𝑒𝑗 ∈ 𝐸 𝐺 − 𝐷. There exists 𝑒𝑖dominates 𝑒𝑗. Definition 2.5[9] A vague set 𝐴 in the universe of discourse 𝑋 is characterized by two membership functions given by
(i) A truth membership function 𝑡𝐴: 𝑋 → [0, 1], (ii) A false membership function𝑓𝐴: 𝑋 → [0, 1].
Where 𝑡𝐴(𝑥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 𝑓𝐴(𝑥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 𝑡𝐴(𝑥)+𝑓𝐴(𝑥) ≤ 1. Thus the grade of membership of x in the vague set 𝐴 is bounded by a subinterval [𝑡𝐴(𝑥), 1 − 𝑓𝐴(𝑥)] of [0, 1].
The vague set 𝐴 is written as 𝐴 = {(𝑥, 𝑡𝐴 𝑥 , 𝑓𝐴 𝑥 )/𝑥 ∈ 𝑋}, where the interval [𝑡𝐴(𝑥), 1 − 𝑓𝐴(𝑥)] is called the value of x in the vague set 𝐴 and denoted by 𝑉𝐴 𝑥 .
Definition 2.6[9] A vague set 𝐴 of a set 𝑋 is called
(i) the zero vague set of 𝑋 if 𝑡𝐴 𝑥 = 0 and 𝑓𝐴 𝑥 = 1 for all 𝑥 ∈ 𝑋, (ii) the unit vague set of 𝑋 if 𝑡𝐴 𝑥 = 1 and 𝑓𝐴 𝑥 = 0 for all 𝑥 ∈ 𝑋.
Definition 2.7 [9]Let 𝐺 = (𝑃, 𝑄) be a vague graph. Then the cardinality of G.vertex cardinality𝑃 and edge cardinality of 𝑄
G = 1 + 𝑡𝑃 𝑣𝑖 − 𝑓𝑃 𝑣𝑖
𝑣𝑖,∈𝑉 2
+ 1 + 𝑡𝑄 𝑣𝑖𝑣𝑗 − 𝑓𝑄(𝑣𝑖𝑣𝑗)
𝑣𝑖𝑣𝑗∈𝐸 2
The vertex cardinality of G is defined by
𝑉 = 1 + 𝑡𝑃 𝑣𝑖 − 𝑓𝑃 𝑣𝑖
𝑣𝑖,∈𝑉 2
; 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉
The edge cardinality of G is defined by
𝐸 = 1 + 𝑡𝑄 𝑣𝑖𝑣𝑗 − 𝑓𝑄(𝑣𝑖𝑣𝑗)
𝑣𝑖𝑣𝑗∈𝐸 2
; 𝑓𝑜𝑟 𝑎𝑙𝑙(𝑣𝑖, 𝑣𝑗) ∈ 𝐸
The number of vertices is called the order of a vague graph and is denoted by 𝑂(𝐺)and the number of edge is called the size of a vague graph and is denoted by 𝑂(𝑆).
Definition 2.8[5]Let 𝐺 = 𝑃, 𝑄 be a vague graph, where 𝑃 = 𝑡𝑃, 𝑓𝑃 is a vague set on 𝑃 ⊆ 𝑄 × 𝑄 such that 𝑡𝑄 𝑥𝑦 ≤ min (𝑡𝑃 𝑥 , 𝑓𝑃 𝑦 and 𝑓𝑄 𝑥𝑦 ≥ max {𝑡𝑃 𝑥 , 𝑓𝑝 𝑦 } for all 𝑥𝑦 ∈ 𝐸.
A vague graph G is said to be strong if 𝑡𝑄 𝑣𝑖𝑣𝑗 = min 𝑡𝑃 𝑣𝑖 , fP(𝑣𝑗) and 𝑓𝑄 𝑣𝑖𝑣𝑗 = max 𝑡𝑃 vi , 𝑓𝑃 𝑣𝑗 for all𝑣𝑖𝑣𝑗 ∈ 𝑄.
Definition 2.9[4]Letu be a vertexin a vague graph𝐺 = (𝑃, 𝑄).Then the neighborhood of 𝑢 is represent by 𝑁 𝑢 = 𝑣 ∈ 𝑉/(𝑢, 𝑣) is a strong arc .
Definition2.10[4]The strong neighborhood of an edge𝑒𝑖 in vague graph 𝐺 = 𝑃, 𝑄 is𝑁𝑠 𝑒𝑖 = 𝑒𝑗 ∈ 𝐸(𝐺) 𝑒𝑗 is a strong arc in 𝐺 and adjecent to 𝑒𝑖 .
Definition 2.11[4]An edge 𝑢, 𝑣 is said to be strong edge in vague graph 𝐺 = 𝑃, 𝑄 if 𝑡𝑄 𝑢𝑣 ≥ 𝑡𝑄 ∞ 𝑢𝑣 and 𝑓𝑄 𝑢𝑣 ≤ (𝑓𝑄)∞ 𝑢𝑣 ,where 𝑡𝑄 ∞ 𝑢𝑣 =
max{ 𝑡𝑄 𝑘 𝑢𝑣 : 𝑘 = 1,2, . . . , 𝑛, and (𝑓𝑄)∞(𝑢𝑣) = min {(𝑓𝑄)𝑘 𝑢𝑣 : 𝑘 = 1,2, . . . , 𝑛}.
Definition 2.12[12]A subset S of V said to be independent set if 𝑡𝑄 𝑢𝑣 < 𝑡𝑄 ∞(𝑢𝑣) and 𝑡𝑄 𝑢𝑣 > 𝑡𝑄 ∞ 𝑢𝑣 .
Definition 2.13[12]Let 𝐺 = 𝑃, 𝑄 be a vague graph, a subset 𝐷 of 𝑄 said to be an edge dominations set in𝐺.If for every in𝑄 − 𝐷 is adjacent to D. The minimum vague graph cardinality of an edge dominating set in G is called the edge domination number of 𝐺 and is denoted by𝛾𝑒,(𝐺).
Definition 2.14[12]Let𝐺 = (𝑃, 𝑄)be a vague graph. For any 𝑢, 𝑣 ∈ 𝑃, we say that u dominate v in 𝐺 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 𝐺 = (𝑃, 𝑄) be avague graphsand 𝑒𝑖, 𝑒𝑗 ∈ 𝑄.Then, we say that 𝑒𝑖dominates𝑒𝑗, if 𝑒𝑖 is a strong arc in 𝐺and adjacent to 𝑒𝑗.
Example 3.2Let𝐺 = (𝑃, 𝑄) be a vague graph, as shown in the figure 3.1. From the edge set 𝑄 = 𝑒1, 𝑒2, 𝑒3 , we have 𝑒1, 𝑒2 are strong arc of 𝑒3.
Definition 3.3Every edge in a vague graph 𝐺 = (𝑃, 𝑄) is called an independent if there is no strong arc between them.
Example 3.4Let𝐺 = (𝑃, 𝑄) be a vague graph, as shown in the figure 3.2. From the edge set 𝑄 = 𝑒1, 𝑒2, 𝑒3, 𝑒4 , we have 𝑒1, 𝑒2, 𝑒4, 𝑒4 there is no strong arc between them.
Figure: 3.2
Definition3.5Let 𝐷 be a minimum dominating set of a vague graph 𝐺.If 𝑒𝑗 ∈ 𝑄 𝐺 − 𝐷, then there exist 𝑒𝑖 ∈ 𝐷 such that 𝑒𝑖 dominates 𝑒𝑗edge.Thus, 𝐷 is called an edge dominating set of 𝐷.
Definition 3.6The minimum cardinality of all edge domination number of vague graph 𝐺 and is denoted by 𝑑𝑒(𝐺)
Example 3.7 Let𝐺 = (𝑃, 𝑄) be a vague graph, as shown in the figure 3.3, we have (𝑒1, 𝑒5) is a minimal edge dominating set in cardinality. Therefore, we get𝑑𝑒 𝐺 =0.60.
Figure:3.3
Note:For any 𝑒𝑖, 𝑒𝑗 ∈ 𝑄 𝐺 . If 𝑒𝑖 dominate 𝑒𝑗 then 𝑒𝑗dominate 𝑒𝑖.Thus, domination is a symmetric relation on 𝑄(𝐺).
e3(0.3,0.5)
Figure: 3.1
𝑒2(0.1,0.7)
𝑣2(0.1,0.6) 𝑒3(0.1,0.5)
𝑣4(0.2,0.3)
𝑒1(0.2,0.6)
𝑣3(0.4,0.5)
𝑣1(0.3,0.5)
𝑒3 0.1,0.8
v1(0.3,0.4) e1(02,0.6)
v2(0.2,0.6)
e2(0.1,0.6) e6(0.3,0.6)
v3(0.2,0.5)
e3(0.1,0.6) v5(0.4,0.5)
e5(0.2,0.6)
e4(0.1,0.6)
v4(0.1,0.6)
Proposition3.8Aedge dominating set 𝐷 of a vague graph𝐺 = (𝑃, 𝑄)is a edge minimal dominating set if and only if forevery𝑒𝑖∈ 𝐷 one of the following conditions hold.
(i) 𝑒𝑖is not a strong neighbor of any edge in 𝐷
(ii) There is a edge 𝑒𝑖∈ 𝑄 − 𝐷 such that N(𝑒𝑖) ∩D = 𝑒 .
Proof.Let Dbe a minimal edge dominating set ofG.Then for every edge𝑒𝑖 ∈ 𝐷,𝐷 – 𝑒𝑖is not a dominatingset and hence there exists an edgee∈Q− (D − 𝑒𝑖 ) which isnot dominated by any edge in𝐷 − 𝑒𝑖 . If 𝑆 = 𝑑, theneis not a strong neighbor of any edge in 𝐷.If 𝐷 − 𝑑 ,then eis not dominated by 𝐷 − 𝑒𝑖 , but itis dominatedby 𝐷. Then the edgeeis a strong neighbor only to din D. That is, 𝑁(𝑒) ∩ 𝐷 = 𝑑.
Conversely, assume that𝐷is a edge dominating set and for each edge𝑑 ∈ 𝐷, one ofthe two conditions holds. Suppose 𝐷is not a minimaldominating set. Then there exists a edge 𝑑 ∈ 𝐷, suchthat 𝐷 − {𝑑}is a edge dominating set. Hence d is a strongneighbor to at least one edgein 𝐷 − {𝑑}, and so (i)does not hold. If𝐷 − {𝑑}is a edge dominating set, then everyedge in𝑄 − 𝐷is a strong neighbor to at least oneedge in𝐷 − {𝑑}, and so (ii) does not hold, which is acontradiction, since at least one of the conditions shouldbe hold. So,𝐷is a minimal dominating set.∎
Proposition 3.9Let 𝐷 be a minimal edge dominating set of connected vague graph 𝐺 = (𝑃, 𝑄).Then,𝐸(𝐺) − 𝐷 is an edge dominating set of 𝐺 = 𝑃, 𝑄 .
Proof.Let 𝐷 is a minimal edge set of vague graph 𝐺 = (𝑃, 𝑄).Since 𝐺 is connected. Then there exist 𝑒𝑖 ∈ 𝑄(𝐺).It is dominated by at least one edge in𝑄(𝐺) − 𝐷 is dominating set.
From the proposition 3.7,It follows that 𝑒𝑗 ∈ 𝑄 𝐺 − 𝐷.Thus every edge in 𝐷 dominated by at least one edge in 𝑄 𝐺 − 𝐷 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.Let𝐷is a maximal edge independent set of 𝐺 = (𝑃, 𝑄).Then𝑒𝑖 ∈ 𝑄 𝐺 − 𝐷.The set 𝐷 ∪ 𝑒𝑖 is not independent for every 𝑒𝑖 ∈ 𝑄 𝐺 − 𝐷. There is anedge 𝑒𝑗 such that 𝑒𝑗 is strong neighbor to 𝑒𝑖 so 𝐷 is a edge dominating set and also edge independent set of 𝐺.
Conversely, if𝐷 is both edge independent and edge dominating set in G. Then,we have to prove 𝐷 is maximal edge independent set having strong edges.Since,𝐷is an edge dominating set having only strong edges, and if𝐷 is not a maximal independent set. Then there exist edge𝑒𝑖not in 𝐷 such that 𝐷 ∪ 𝑒𝑖 is a dominating set, then no any edge in 𝐷 is strong neighbor to 𝑒𝑖. Hence,𝐷cannot be a dominating set, which is a contradiction.
Thus,𝐷 is maximal edge independent set of𝐺 = 𝑃, 𝑄) having only strong edges. ∎
Proposition 3.11Every maximal edge independent set in a vague graph 𝐺 = (𝑃, 𝑄)having only strong edges is a minimal edge dominating set of 𝐺 = 𝑃, 𝑄 .
Proof.Let 𝐷 be a maximal edge independent set having only strong edge of an vague graph 𝐺(From the proposition 3.9) is a edge dominating set of 𝐺.
Assume 𝐷 is not minimal edge dominating set. Then there exist atleast one edge 𝑒𝑖in 𝐷. such that 𝐷 − 𝑒𝑖 dominates 𝑄 𝐺 − 𝐷 − 𝑒𝑖 is an 𝑁𝑠 𝑒𝑖 which is an contradiction. Therefore,𝐷 is an independent set of 𝐺. Hence, 𝐷 is must be minimal dominating set. ∎
Proposition 3.12If 𝐷 is an edge dominating set of a vague graph 𝐺 which is containing at least one dominating set in G.
Proof.Let D be an edge dominating set of vague graph 𝐺. Suppose edge dominating set 𝐷 of a vague graph 𝐺 no dominating set 𝐷’in 𝐺.From the proposition 3.8, if any two nodes of 𝐷 are independent and non-adjacent. Then,𝑒𝑖 not in strong neighborhood of 𝑒𝑗.Therefore, for every 𝑒𝑖 ∈ 𝑄 − 𝐷.There exist no 𝑒𝑖 in 𝐷,such that 𝑒𝑖 will not dominating 𝑒𝑗. Which is a contraction to the edge domination set 𝐷. Hence, we have 𝐷 must contain at least one dominating set 𝐷’in 𝐺. ∎
Example 3.14 Let𝐺 = (𝑃, 𝑄)be a vague graph, as shown in the figure 3.4.From 𝐷 = 𝑒3,𝑒5 is an edge dominating set, Then 𝐸 − 𝐷 = 𝑒1, 𝑒2, 𝑒4, 𝑒6 ,Thus, 𝑒3,𝑒5 = 𝑣1, 𝑣3, 𝑣4, 𝑣5 .Let 𝐷’= 𝑣1, 𝑣5 then 𝑉 − 𝐷′ = 𝑣3, 𝑣5 Therefore 𝐷′ is dominating set.
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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Figure:3.4
(0.2,0.7) (0.1,0.5) v1(0.2,0.4)
e2(0.2,0.5)
v2(o.3,0.4)
e3(0.2,0.5)
v3(0.2,0.5)
e4(0.1,0.6)
v4(0.4,0.5) e5(0.3,0.6)
v5(0.3,0.5) e6(0.1,0.6) v6(0.3,0.6)
e1(0.2,0.6)
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