The Cycle Non Split Domination Number of An Intuitionistic Fuzzy Graph

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1 9 The Cycle Non Split Domination Number of An Intuitionistic Fuzzy Graph Ponnappan C. Y., Department of Mathematics, Government Arts College Paramakudi, Tamilnadu, India Surulinathan P., Department of Mathematics, Lathamathavan Engineering College, Kidaripatti, Alogarkovil, Madurai Basheer Ahamed S., Department of Mathematics, P.S.N.A. College of Engineering and Technology, Dindigul ABSTRACT (V, E) is a cycle non split dominating set if the induced intuitionistic fuzzy subgraph H = (<V - D>, V', E')is a cycle. The cycle non split domination number γ cns (G) of G is the minimum intuitionistic fuzzy cardinality of a cycle non spilt dominating set. In this paper we study a cycle non split dominating sets of an intuitionistic fuzzy graphs and investigate the relationship of γ cns (G) with other known parameters of G. Keywords: Intuitionsitic Fuzzy graph, Intuitionistic fuzzy domination, spilt domination number in IFG, Non split domination in IFG, cycle non split domination number in IFG. AMS classification Number :05C. INTRODUCTION The study of domination set in graphs was begun by ore and Berge. Kulli V.R et.al [2] introduced the concept of split domination and non split domination in graphs. Rosenfield[9] introduced the notion of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as path, cycles and connectedness. A. Somasundram and S.Somasundaram[0] discussed domination in fuzzy graphs. Q.M. Mahyoub and N.D. Sonar [3] discussed the split domination number of fuzzy graphs C.Y Ponnappan et.al [8] discussed the strong non split domination number of fuzzy graphs. K.T. Atanassov[] initiated the concept of intuitionistic fuzzy relations and intuitionitic fuzzy graphs. R.Parvathi and M.G. Karunambigai[6] gave a definition of IFG as a special case of IFGS defined by K.T Atanassov and A.Shannon. R.Parvathi and G.Thamizhendhi[7] was introduced dominating set and domination number in IFGS. NagoorGani. A andanupriya.s[4] was introduced the split domination on intuitionistic fuzzy graphs. In this paper, we discuss the cycle non split domination number of an intuitionstic fuzzy graphs and obtain the relationship with other known parameters of an IFG G. 2. PRELIMINARIES Definition: 2. An Intuitionistic Fuzzy Graph (IFG) is of the form G =(V, E) where (i) V = {V, V 2,, V n } such that σ : V [0, ] andσ 2 : V [0, ] denote the degree of membership and non-membership of the element v i V respectively and 0 σ (v i ) + σ 2 (v i ) =, for every v i V (ii) E V x V where µ :V x V [0, ] and µ 2 : V x V [0, ] are such that µ (v i, v j ) min {σ (v i ), σ (v j )} µ 2 (v i, v j ) max {σ 2 (v i ), σ 2 (v j )} and0 µ (v i, v j ) + µ 2 (v i, v j ) for every (v i, v j ) E. Note : when µ ij = µ 2ij =0 for some i and j then there is no edge between v i and v j otherwise there exits an edge between v i and v j. Definition: 2.2 An IFG H = (V', E') is said to be an IF subgraph (IFSG) of G = (V, E) if V' E and E' E. That is σ i σ i ; σ σ and µ i µ i ; µ µ for everyi =, 2.n 2i 2i 2j 2j, Definition: 2.3 The intuitionistic fuzzy subgraph H = (V, E ) is said to be a spanning fuzzy subgraph of an IFG G = (V, E) if σ (u) =σ(u) and σ 2 (u) =σ2(u) for all u V and µ (u,v) µ (u,v) and µ 2(u, v) µ 2(u, v) for all u,v V. Definition: 2.4 Let G = (V, E) be an IFG. Then the vertex cardinality of G is defined by p = V = ( +σ( v) σ2( v2) ) for all vi V 2 vi v Definition: 2.5 Let G = (V, E) be an IFG. Then the edge cardinality of E is defined by

2 0 vv i j E ( ( i j) 2( i j) ) ( i j) q = E = v,v v,v forall v,v E 2 +µ µ Definition: 2.6 Let G = (V, E) be an IFG. Then the cardinality of G is defined to be G = V + E = p +q Definition: 2.7 The number of vertices is called the order of an IFG and is denoted by O(G). The number of edge is called size of IFG and is denoted by S(G). Definition: 2.8 The vertices v i and v j are said to the neighbors in IFG either one of the following conditions hold (i) µ (v i, v j ) > 0, µ 2 (v i, v j ) > 0 (ii) µ (v i, v j ) = 0, µ 2 (v i, v j ) > 0 (iii) µ (v i, v j ) > 0, µ 2 (v i, v j ) = 0, v i, v j V Definition: 2.9 A path in an IFG is a sequence of distinct vertices v, v 2, v n such that either one of the following conditions is satisfied. (i) µ (v i, v j ) > 0, µ 2 (v i, v j ) > 0 for some i and j (ii) µ (v i, v j ) = 0, µ 2 (v i, v j ) > 0 for some i and j (iii) µ (v i, v j ) > 0, µ 2 (v i, v j ) = 0 for some i and j Note : The length of the path P = v, v 2, v n+ (n > 0) is n Definition: 2.0 Two vertices that are joined by a path is called connected. Definition: 2. An IFG G = (V, E) is said to be complete IFG if µ ij = min {σ i,σ j } and µ 2ij = max {σ 2i, σ 2j } for every v i, v j V. Definition: 2.2 The complement of an IFG, G = (V, E) is an G = V, E, where IFG, ( ) (i) V = V (ii) σ i =σi and σ 2i =σ 2i, for all i =, 2, n µ = min σ, σ µ (iii) { } (iv) max {, } ij i j ij µ = σ σ µ for alli =, 2, n 2ij 2i 2j 2ij Definition: 2.3 An IFG, G = (V, E) is said to bipartite the vertex set V can be partitioned into two non empty sets V and V 2 such that (i) µ (v i, v j ) = 0 andµ 2 (v i, v j ) = 0 if v i, v j V (or) v i, v j V 2 (ii) µ (v i, v j ) > 0 and µ 2 (v i, v j )> 0 if v i V and v j V 2 for some i and j or µ (v i, v j ) = 0 and µ 2 (v i, v j ) > 0 if v i V and v j V 2 for some i and j or µ (v i, v j ) > 0 and µ 2 (v i, v j ) = 0 if v i V and v j V 2 for some i and j Definition: 2.4 A bipartite IFG, G = (V, E) is said to be complete if µ (v i, v j ) = min {σ (v i ), σ (v j )} µ 2 (v i, v j ) = max {σ 2 (v i ), σ 2 (v j )} forall v i V and v j V 2. It is denoted by K ( σσ, µµ ) 2 2 Definition: 2.5 A vertex u V of an IFG G = (V, E) is said to be an isolated vertex if µ (u, v) = 0 and µ 2 (u, v) = 0, for all v V. That is N (u) = φ. Thus an isolated vertex does not dominate any other vertex in G. Definition: 2.6 Let G = (V, E) be an IFG on V. Let u, v V, we say that u dominate v in G if there exits a strong arc between them. Definition: 2.7 A subset D of V is called a dominating set in IFG G if for every v V D, there exists u D such that u dominates v. Definition: 2.8 A dominating set D of an IFG is said to be minimal dominating set if no proper subset of D is a dominating set. Definition: 2.9 A vertex v G is said to be end vertex of IFG if it has almost one strong neighbor in G. Definition: 2.20 (V, E) is aintuitionistic split dominating set if the induced fuzzy subgraphh = (<V - D>, V', E') is disconnected. The split domination number γ s (G) of G is the minimum intuitionistic fuzzy cardinality of a split domination set. Definition: 2.2 (V, E) is aintuitionistic fuzzy graph G = (V, E) is a non split dominating set if the induced intuitionistic fuzzy sub graph H = (<V - D>,V', E') is connected. The non split domination number γ ns (D) of G is the minimum fuzzy cardinality of a non split domination set.

3 Definition: 2.22 A dominating set D of aintuitionistic fuzzy graph G = (V, E) is a cycle non split dominating set if the induced fuzzy subgraph H = (<V - D>,V', E') isa cycle. The cycle non split domination number γ cns (G) of G is the minimum intuitionistic fuzzy cardinality of a cycle non split domination set. Definition: 2.23 A dominating set D of a intuitionistic fuzzy graph G = (V, E) is a path non split dominating set if the induced fuzzy sub graph H = (<V - D>,V', E') is a path. The path non split domination number γ pns (G) of G is the minimum intuitionistic fuzzy cardinality of a path non split domination set. 3. MAIN RESULTS Theorem:3. For any fuzzy intuitionistic fuzzy G = V,E, γ G γ G p graph ( ) ( ) ( ) Example : (0.,0.2) D={V 2, V 5 } γ(g) = 0.85 D cns = {V,V 6 } γ cns (G) = 0.9 γ(g) γ cns (G) cns v 2 v 5 (0.,0.3) v v 6 (0.4,0.2) v 3 (0.3, 0.2) v 4 Fig (i) (0.3,0.) Let G =(V,E) be an intuitionistic fuzzy graph. Let D be the minimum intuitionistic fuzzy dominating set. D cns is the intuitionistic fuzzy cycle non split dominating set. D cns is also a dominating set but need not be minimum intuitionistic fuzzy dominating set. Therefore D D cns ie γ(g) γ cns (G) Clearly γ(g) γ cns (G) p (0.,0.2) u Fig (ii) γ(g) =0.85, γ s (G) =.45, γ cns (G) = 0.9 γ(g) min {γ s (G), γ cns (G)} Let G = (V, E) be an intuitionistic fuzzy graph. Let D be the minimum intuitionistic fuzzy dominating set. Let D s and D cns be the minimum intuitionistic fuzzy split dominating set and minimum intuitionistic fuzzy cycle non split dominating set of G respectively. The cardinality of intuitionistic fuzzy dominating set need not exceeds either one of the minimum cardinality of intuitionistic fuzzy split dominating set or intuitionistic fuzzy cycle non split dominating set. Theorem 3.2. γ(g) min {γ s (G), γ pns (G)} Theorem: 3.3 For any intuitionistic spanning fuzzy sub graph H = (V, E ) of IFG G = (V, E),γ cns (H) γ cns (G). Example :From Fig.(i) H: (V, E ) : (0.3,0.5) u 5 v (0.,0.5) v 2 (0.,0.2) (0.,0.4) u 2 (0.2,0.4) (0.2,0.5) (0.3,0.5) u 3 (0.3,0.5) (0.3,0.5) v 5 v 3 (0.3, 0.2) v 4 (0.4, 0.2) (0.3, 0.) u 4 (0.5,0.4) v 6 Theorem 3.. γ(g) γ pns (G) p Theorem: 3.2 For any intuitionistic fuzzy graph G = (V,E), γ(g) min {γ s (G), γ cns (G)}. γ cns (G) = 0.9 D cns (H) = {v, v 5, v 6 } γ cns (H) =.3 Fig (iii)

4 2.3 = γ cns (H) γ cns (G) = 0.9 Let G = (V,E) be an intuitionistic fuzzy graph and let H = (V, E ) be the intuitionistic spanning fuzzy sub graph of G. D cns (G) be the intuitionistic fuzzy minimum cycle non split dominating set of G. D cns (H) be the intuitionistic fuzzy cycle non split dominating set of H but not minimum. γ cns (H) γ cns (G). Theorem: 3.3.γ pns (H) γ pns (G). Theorem: 3.4 For any intuitionistic complete fuzzy graph K ( σσ ) 2, µµ 2 γ(g) = γ cns (G) = min { v i / v i v} Let G = (V, E) be a complete fuzzy graph therefore, there is a strong arcbetween every pair of vertices. We remove any vertex having minimum cardinality, the resulting graph is a cycle. Example : (0., 0.4) v v 2 (0.3, 0.4) (0.3, 0.4) any element of D. If u = v we get (i) and if u v we get (ii) The converse is obvious. Theorem: 3.5. γ pns - set satisfies Ore s theorem. Theorem: 3.6 Let G be a complete intuitionistic fuzzy graph K ( σσ 2, µµ 2) then γ cns (G) = min { u }, where u is the vertex having minimum intuitionistic fuzzy cardinality. Let G i bea sub graph of G induced by<v-u> where u is the vertex of minimum cardinality, G i has a vertex set V i = {V-u} then γ cns (G) γ cns (G ) γ cns (G 2 ) γ cns (G n ) provided the intuitionistic fuzzy graph G n is a elementary cycle with three vertices. Theorem: 3.7 For any fuzzy graph without isolated vertices p/2. v (0.2,0.3 ) (0.,0.3) v 2 (0.5,0.2) (0.,0.3) V 5 (0.,0.) (0.4,0.6) (0.2, 0.4) V 4 ( 0.3,0.3) (0.3,0.6) V 3(0.4,0.6) (0., 0.6) (0.3, 0.2) v 3 v 4 Fig. (iv) Let D = {v} is minimum dominating set, then <V D> is a cycle. γ(g) = γ cns (G) = min { v i / v i v}. Theorem: 3.5 In an intuitionistic fuzzy graph G. D cns is the minimal cycle non split dominating set iff for each v D cns, one of the following two conditions holds. (i) N(v) D cns = φ (ii) There is a vertex u V D cns such thatn(u) D cns = {v}. Let D cns be a minimal cycle non split domination set of an intuitionistic fuzzy graph and v D cns, then D = D cns {v} is not a cycle non split dominating set and hence there exist u V - D such that u is not dominated by Fig.(v) Proof : Any graph without isolated vertices has two disjoint dominating sets and hence the result follows. Theorem: 3.8 For any intuitionistic fuzzy graph, cns (G) p - E From fig. (iv) p = 2.85, E =.3, cns (G)=0.8 V 6(0.,0.4) Let v be a vertex of a intuitionistic fuzzy graph, such that dn(v) = E, then V\N(v) is a dominating set of G, so that γ cns (G) V \ N(v) = p E. Theorem: 3.9 For the domination number γ cns the following theorem gives a Nordhaus Gaddum type result.

5 3 For any intuitionistic fuzzy graph G, γ cns (G) + γ cns ( G ) 2p. Let G be a connected intuitionistic fuzzy graph it may or may not contain a cycle. [0] Somasundaram,A and Somasundaram,S., Domination in Fuzzy Graphs-I, Pattern Recognition letters, (998), Suppose G contains a cycle then by theorem 3., γ cns (G) p. Also G may or may not contain a cycle. We have γ cns ( G ) p or γ cns ( G ) = 0Vice versa. Hence the inequality is trivial. Theorem 3.9 γ pns (G) + γ pns ( G ) 2p. REFERENCES [] Atanassov K.T., Shannon A., On a Generalization of Intuitionistic Fuzzy Graphs, NIFS Vol. 2,, 24-29, [2] Kulli, V.R., Theory of domination in Graphs Vishwa International Publications 202. [3] Mahioub, Q.M., and Soner, N.D., The split domination number of fuzzy graphs, Far East journal of Applied Mathematics volume 30, Issue-, pages (Jan 2008). [4] NagoorGani. A., and Anupriya. S., Split Domination on Intuitionistic Fuzzy Graph, Advance in Computational Mathematics and its Applications (ACMA), Vol.2, No.2, 202, ISSN [5] Ore,O. (962). Theory of Graphs. American Mathematical Society Colloq.Publi., Providence, RI, 38. [6] Parvathi, R., and Karunambigai, M.G., Intuitionistic Fuzzy Graphs, Computational Intelligence, Theory and applications, International Conference in Germany, Sept 8-20, [7] Parvathi, R., and Tamizhendhi, G., Domination in intuitionistic fuzzy graphs, Fourteen Int. Conf. on IESs, Sofia, NIFS Vol. 6, 2, 39-49, 5-6 May 200. [8] Ponnappan C.Y, BasheerAhamed.S.,Surulinathan.P, The strong split domination number of fuzzy graphs International Journal of Computer & Organization Trends- Volume 8 Number 2 May 204. ISSN: [9] Rosenfeld A. Fuzzy graphs. In LA. Zadeh, KS. Fu, M.Shimura (eds) Fuzzy sets and their applications, Academic, NewYork, 975