Global Domination Set in Intuitionistic Fuzzy Graph

ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 9 || September – 2014 ||
International Journal of Computational Engineering Research (IJCER)
www.ijceronline.com Open Access Journal Page 55
Global Domination Set in Intuitionistic Fuzzy Graph R. JahirHussain 1 ,S. Yahya Mohamed 2 1P.G and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli-620 020, India 2P.G and Research Department of Mathematics, Govt. Arts College, Trichy-22, India
I. INTRODUCTION
Atanassov [1] introduced the concept of intuitionistic fuzzy (IF) relations and intuitionistic fuzzy graphs (IFGs). Research on the theory of intuitionistic fuzzy sets (IFSs) has been witnessing an exponential growth in Mathematics and its applications. R. Parvathy and M.G.Karunambigai’s paper [7] introduced the concept of IFG and analyzed its components. Nagoor Gani, A and Sajitha Begum, S [5] defined degree, Order and Size in intuitionistic fuzzy graphs and extend the properties. The concept of Domination in fuzzy graphs is introduced by A. Somasundaram and S. Somasundaram [8] in the year 1998. Parvathi and Thamizhendhi[6] introduced the concepts of domination number in Intuitionistic fuzzy graphs. Study on domination concepts in Intuitionistic fuzzy graphs are more convenient than fuzzy graphs, which is useful in the traffic density and telecommunication systems. The Global domination number of a Graph was discussed by E. Sampathkumar [10] in 1989.In this paper, We define global Intuitionistic fuzzy domination set of IFG and discuss the situation of this concept used in network. Also some thereoms and bounds of global Intuitionistic fuzzy domination number of IFGs are established. II. PRELIMINARIES Definition 2.1: An Intuitionistic fuzzy graph is of the form G = (V, E ) where (i) V={v1,v2,….,vn} such that μ1: V[0,1]and γ1: V [0,1] denote the degree of membership and non-membership of the element vi V, respectively, and 0 ≤ μ1 (vi) + γ1 (vi) ≤ 1 for every vi V, (i = 1,2, ……. n), (ii) E V x V where μ2: VxV [0,1] and γ2: VxV [0,1] are such that μ2 (vi , vj) ≤ min [μ1(vi), μ1(vj)] and γ2 (vi , vj) ≤ max [γ1(vi), γ1(vj) ] and 0 ≤ μ2 (vi, vj) + γ2 (vi, vj) ≤ 1 for every (vi ,vj) E, ( i, j = 1,2, ……. n) Definition 2.2 An IFG H = < V’, E’ > is said to be an Intuitionistic fuzzy subgraph (IFSG) of the IFG, G = < V, E > if V’ V and E’ ⊆ E. In other words, if μ1i’ ≤ μ1i ; γ1i’ ≥ γ1i and μ2ij’ ≤ μ2ij ; γ2ij’ ≥ γ2ij for every i, j = 1,2………n. Definition 2.3: Let G = (V,E) be a IFG. Then the cardinality of G is defined as |G| = Definition 2.4: The vertex cardinality of IFG G is defined by |V| = = p and The edge cardinality of IFG G is defined by |E| = = q. The vertex cardinality of IFG is called the order of G and denoted by O(G). The cardinality of G is called the size of G, denoted by S(G).
ABSTRACT
In this paper, We define global Intuitionistic fuzzy domination set and its number of IFGs. Also connected Intuitionistic fuzzy domination number of IFGs are discussed. Some results and bounds of global Intutionistic fuzzy domination number of IFGs are established.
KEYWORDS: Intuitionistic fuzzy graph, connected Intuitionistic fuzzy dominating set, global Intuitionistic fuzzy dominating set, effective degree.
2010Mathematics Subject Classification: 05C69, 03F55, 05C72, 03E72.

Global Domination Set In …
Definition 2.5: An edge e = (x, y) of an IFG G = ( V, E ) is called an effective edge if μ2(x, y) = μ1(x) Λ μ1 (y) and γ2(x, y) = γ1(x) V γ1(y). Definition 2.6: An Intuitionistic fuzzy graph is complete if μ2ij = min ( μ1i, μ1j ) and γ2ij = max (γ2i ,γ2j) for all ( vi , vj ) V. Definition 2.7: An Intuitionistic fuzzy graph G is said to be strong IFG if μ2(x, y) = μ1(x) Λ μ1 (y) and γ2(x, y) = γ1(x) V γ1(y) for all ( vi , vj ) E. That is every edge is effective edge. Definition 2.8 : The complement of an IFG G = < V, E > is denoted by = ( , ) and is defined as i) = μ1(v) and = γ1(v) ii) (u,v) = μ1 (u) Λ μ1 (v) - μ2 (u,v) and (u,v) = γ1(u) V γ1(v) - γ2(u,v) for u,v in V Definition 2.9: Let G = (V,E) be an IFG. The neighbourhood of any vertex v is defined as N(v) = (Nμ(v) , Nγ(v)), Where Nμ(v) = and Nγ(v) = . N[v] = N (v) {v} is called the closed neighbourhood of v. Definition 2.10: The neighbourhood degree of a vertex is defined as dN(v) = (dNμ(v),dNγ(v)) where dNμ(v) = and dNγ(v) = . The minimum neighbourhood degree is defined as N(G) = (Nμ(v),Nγ(v)), where Nμ(v) = { dNμ(v): vV} and Nγ(v) = { dNγ(v): vV}. Definition 2.11: The effective degree of a vertex v in a IFG. G = (V, E) is defined to be sum of the effective edges incident at v, and denoted by dE(v). The minimum effective degree of G is E(G) = Λ {dE (v)/v V} Definition 2.12: Let G= (V, E) be an IFG. Let u, v V, we say that u dominated v in G if there exist a strong arc between them. A Intuitionistic fuzzy subset D V is said to be dominating set in G if for every v V-D, there exist u in D such that u dominated v. The minimum scalar cardinality taken over all Intuitionisitc fuzzy dominating sets is called Intuitionistic fuzzy domination number and is denoted by γ. The maximum scalar cardinality of a minimal domination set is called upper Intuitionistic fuzzy domination number and is denoted by the symbol Γ . Definition 2.13: A Intuitionistic fuzzy dominating set D⊆V of IFG G is said to be a Intutitionisitc fuzzy connected dominating set of G if the subgraph <D> induced by D is connected. The minimum cardinality taken over all minimal Intuitionistic fuzzy connected dominating sets is called Intuitionistic fuzzy domination number of G and it is denoted by γc(G). Definition 2.14: An independent set of an Intuitionistic fuzzy graph G = (V, E) is a subset S of V such that no two vertices of S are adjacent in G. Definition 2.15: A Bipartite IFG, G= (V,E) is said to be complete Bipartite IFG, if μ2(vi, vj) = μ1(vi) Λ μ1 (vj) and γ2(vi, vj) = γ1(vi) V γ1(vj) for all vi V1 and vj V2. It is denoted by . III. GLOBAL INTUITIONISTIC FUZZY DOMINATION SET IN IFG Definition 3.1: Let G = (V, E) be an IFG. A Intuitionistic fuzzy dominating set S ⊆V is said to be global Intuitionistic fuzzy dominating set of G if S is also a Intuitionistic fuzzy dominating set of . The minimum cardinality of global Intuitionistic fuzzy dominating sets is global Intuitionistic fuzzy domination number and is denoted by γg(G). Example: In case of transportation and road networks, the travel time is mostly used as weight. The travel time is a function of the traffic density on the road and/or the length of the road. The length of a road is a crisp quantity but the traffic density is fuzzy. In a road network, we represent crossings as nodes and roads as edges. The traffic density is mostly calculated on the road between adjacent crossings. These numbers can be represented as intuitionistic fuzzy numbers. Road network represented as an intuitionistic fuzzy graph R* = (C, L), where C is an intuitionistic fuzzy set of crossings at which the traffic density is calculated and L is an intuitionistic fuzzy set of roads between two crossings. The degrees of membership, μL(xy), and non membership, νL(xy), are calculated as μL(xy) = min (μC(x), μC(y)), γL(xy) = max (γC(x), γC(y)).

Some essential goods are being supplied to some crossings from supplying stations located some other crossings. It may happen that the roads(edges of G) may be closed for some reason or the other. So, we have to think of maintaining the supply of goods to various crossing uninterrupted through secret links ( i.e. edges of the complement of network). We have to find minimum number of supplying stations(crossings) needed which is called global Intuitionistic fuzzy domination number. Example 3.2: Let G = (V,E) be IFG be defined as follows
v1(0.2, 0.4) (0.2, 0.4) v2(0.3,0.3) (0.2,0.6) (0.3,0.6) (0.1,0.5) v4(0.4,0.6) (0.1, 0.6) V3(0.1,0.5) Fig- 1: Intuitionistic fuzzy graph(G) Here |v1| = 0.4 , |v2| = 0.5, |v3| = 0.3, |v4| = 0.4 and minimum γg-set is { v2, v3, v4} and therefore γg(G) = 1.2. Observations 3.3:
(i) γg(Kn) = γg() = p
(ii) γg() = Min {|vi|} + Min {|vj|}, where vi V1 and vj V2.
Proposition 3.4: The global Intuitionistic fuzzy dominating set is not singleton. Proof: Since gifd-set contain dominating set for both G and Gsc then at least two vertices are in the set. i.e) The gifd-set containing at least two vertices. Theorem 3.5: For any IFG G= (V, E) with effective edges, Min{|Vi|+|Vj|} ≤ γg(G) ≤ p, i ≠ j Proof: We know that global Intuitonistic fuzzy dominating set has at least two vertices. Let {vi, vj} are the vertices, then Min{|Vi|+|Vj|} = γg(G) If the set contains other than {vi, vj} then Min{|Vi|+|Vj|} < γsg(G), i ≠ j If the given G is complete IFG then gifd-set contains all the vertices of the G, that is γg(G) ≤ O(G) = p i.e.) We get, Min{|Vi|+|Vj|} ≤ γg(G) ≤ p. Theorem 3.6: Let G = (V, E) be the IFG and the Intuitionistic fuzzy dominating set S of G is global Intuitionistic fuzzy dominating set if and only if, for each v V-S, there exists a u S such that u is not adjacent to v. Proof: Let S is global dominating set and also dominating set. Suppose u is adjacent to v then we get S is not a dominating set. Which is contradiction. That is u is not adjacent to v. Conversely, for each v V-S and u is not adjacent to v then the set S is dominating both G and . That is S is global Intuitionistic fuzzy dominating set. Theorem 3.7: Let G = (V, E) be an IFG then (i) γg(G) = γg() (ii) γ (G) ≤ γg(G) Proof: G is connected IFG and γg-set dominating vertices of G and . Clearly γg(G) = γg() . Suppose D is the γ-set of G then the number of vertices in the dominating set is less than or some time equal to γg-set. That is γ (G) ≤ γg(G). Theorem 3.8: Let S be the minimum Intuitionisitc fuzzy dominating set of IFG G containing t vertices. If there exist a vertex v V-S adjacent to only vertices in S then γg-set contain atmost t+1 vertices. Proof: Since S is the γ-set and v V-S adjacent to only vertices in S then we get S{v} is a global Intuitionistics fuzzy dominating set. That is γg-set contain atmost t+1 vertices. Theorem 3.9: Let G = (V, E) be strongly connected IFG then, at least one of the following holds.
(i) γc(G) ≤ γg(G). (ii) (G) ≤ γg(G).
Proof: Since γc-set also dominating set and induced Intuitionistic fuzzy subgraph is connected then the Gc may be disconnected and it is less than or equal to γg-set. That is γc(G) ≤ γg(G). Similarly, we have (G) ≤ γg(G).

Definition 3.10: G = (V, E ) be a connected IFG with effective edges which is said to be semi complete IFG, if every pair of vertices have a common neighbor in G . The IFG G is said to be purely semi complete IFG if G is semi complete IFG but not complete IFG. Example 3.11:
v1(0.2, 0.3) (0.2, 0.4) v2(0.3,0.4) (0.2,0.5) (0.2,0.6) (0.3,0.6) v4(0.4,0.5) (0.3, 0.6) v3(0.3,0.6) Fig-2: Purely Semi complete IFG Theorem 3.12: Let G =(V, E) be the purely semi complete IFG. Then γg-set contains at least three vertices. Proof: Since G is purely semi complete IFG then it contains triangles with common vertex. Let v be the common vertex then, In Gc , the vertex v is isolated vertex. Also global Intuitionistic dominating set contains Intuitionistic fuzzy dominating vertices of G and Gc. Suppose gifd-set contains less than three vertices We know that gifd-set not a singleton. i.e) gifd-set contains at least two vertices Let D = { v1, v2} be a gifd-set in G. Case 1: <D> is connected in G Then v1v2 is an effective edge in G. By the definition of semi complete IFG, there is a v3 in G such that <v1v2v3> is triangle in G, i.e) D is not a Intuitonistic fuzzy domination set in Gc. Which is contradiction to D is a gifd-set in G. Case 2: <D> is disconnected in G. i.e.) There is no effective edge between v1 and v2 . Since G is semi complete IFG, there is v3 in G such that v1v3and v3v2 are the effective edges in G Therefore, In Gc, v3 is not dominated by a vertex in D. Which implies, D is not a gifd-set in G Which is contradiction to our assumption. That is γg-set contains at least three vertices Theorem 3.13: Let G = (V, E) be the IFG with effective edges . γg(G) = Min{|Vi|+|Vj|} i ≠j if and only if there is an effective edge uv in G such that each vertex in V – {u, v} is adjacent to u or v but not both. Proof: Suppose γg(G) = Min{|Vi|+|Vj|} i≠j, We assume D = {u, v} be the gifd-set in G Let <D> is connected in G, then uv is an effective edge in G. If any vertex w in V-{u, v} is adjacent to both u and v. Which implies D is not a dominating set for Gc. which is contradiction to our assumption. i.e) effective edge uv in G such that each vertex in V – {u, v} is adjacent to u or v but not both. Conversely, each vertex in V – {u, v} is adjacent to u or v but not both, then we get γg(G) = Min{|Vi|+|Vj|} i≠j. IV. CONCLUSION Here ,We defined global Intuitionistic fuzzy domination set of IFG and discussed the situation of this concept used in network. Also some theorems and bounds of global Intuitionistic fuzzy domination number of IFGs are established. Further we going to establish more results and bounds on this gifd number with other domination parameters. REFERENCES
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