Nil Complementary Domination in Intutionistic Fuzzy Graph

1. International Journal of Trend in International Open Access Journal International Open Access Journal | www.ijtsrd.com International Journal of Trend in Scientific Research and Development (IJTSRD) Research and Development (IJTSRD) www.ijtsrd.com ISSN No: 2456 ISSN No: 2456 - 6470 | Volume - 2 | Issue – 6 | Sep 6 | Sep – Oct 2018 Nil Complementary Domination in Intutionistic Fuzzy Graph Complementary Domination in Intutionistic Fuzzy Graph Complementary Domination in Intutionistic Fuzzy Graph R. Buvaneswari1, K. Jeyadurga2 R. 1Assistant Professor, Assistant Professor, 2Research Scholar KG College of Arts and Science, Coimbatore, Tamil Nadu, India KG College of Arts and Science, Coimbatore, Tamil Nadu , India ABSTRACT The author defines the complementary dominating set and its number in intutionistic fuzzy graph. The author discussed the order and enclave is obtained for some standard intutionistic fuzzy graph is derived. Keyword: Intutionistic complementary nil dominating set, effective degree. 1.INTRODUCTION In 1999, Atanassov[1] introduced the concept of intutionistic fuzzy relation and intutionistic fuzzy graphs. Parvathi and Karunambigai [5] introduced the concept of intutionistic fuzzy graph and analyzed its components. Nagoor Gani and Sajitha Begum [11] define degree, order and size in intutionistic fuzzy graph and derive its some of their properties. Somasundaram [9] introduced the concept of domination in intutionistic fuzzy graph. P Thamizhendhi [6] introduced the concept of domination number in intutionistic fuzzy graph. Tamizh Chelvam [10] introduced and analyzed the complementary nil dominating set in the crisp graph. The properties of nil complementary dominating graph in intutionistic fuzzy graph is discussed in this paper. 2. Preliminaries Definition 2.1 A fuzzy graph ( µ σ, = G [ ] 1 , 0 : → V σ and : → ×V V µ ( u v u σ σ µ ∧ = , . Definition 2.2 Let ( E V G , = be an intutionistic fuzzy graph, such that (i) { } v v V ...... , 2 1 = such that µ [ ] 1 , 0 : 1 → V ν denote the degree of membership denote the degree of membership and non-membership of the element respectively and 0 ≤ , V vi∈ (i=1,2,......n). V V E × ⊆ where [ ] 1 , 0 : 2 → ×V V ν are ( i j i v v v 1 2 , µ µ µ ∧ ≤ ( i j i v v v 1 2 , ν ν ν ∧ ≤ ( , 0 2 2 + ≤ i j i v v v ν µ The author defines the complementary dominating set and its number in intutionistic fuzzy graph. The author discussed the order and enclave is obtained for some standard intutionistic fuzzy graph is derived. vi∈ for every membership of the element V ( ) ( ) v µ + ν ≤ 1 v v 1 1 i i [ ] 1 , 0 µ ×V → (ii) : V and that and and 2 Intutionistic fuzzy fuzzy graph, graph, de degree, are such such ) ) ( ) v ( ) j v ) j v ) complementary nil dominating set, effective degree. ( ) ( ) 1 j respectively 1 ) ( Atanassov[1] introduced the concept of intutionistic fuzzy relation and intutionistic fuzzy graphs. Parvathi and Karunambigai [5] introduced the ≤ ≤ , 1 Definition 2.3 Let G = cardinality of G is defined to be ∑ ∈ V v i graph and analyzed its ( ) , V E , be an intutionistic fuzzy graph. The , be an intutionistic fuzzy graph. The ality of G is defined to be ∑ v components. Nagoor Gani and Sajitha Begum [11] define degree, order and size in intutionistic fuzzy graph and derive its some of their properties. Somasundaram [9] introduced the concept of domination in intutionistic fuzzy graph. Parvathi and Thamizhendhi [6] introduced the concept of domination number in intutionistic fuzzy graph. Tamizh Chelvam [10] introduced and analyzed the complementary nil dominating set in the crisp graph. The properties of nil complementary dominating in intutionistic fuzzy graph is discussed in this ( ) ij e ( ) ij e ( ) v ( ) v + µ − ν 1 + µ − ν 1 2 2 = + 1 1 i i G 2 2 ∈ V i The vertex cardinality is defined as The vertex cardinality is defined as ( ) ( ) v + µ − ν 1 v v ∑ ∈ vi = 1 1 i i V 2 V The edge cardinality is defined The edge cardinality is defined as ( ) ( ) ij e + µ − ν 1 e e ∑ ∈ eij 2 2 ij = E 2 V Definition 2.4 Let G = degree ( ) i v d = ( ) , ) [ ] 1 , 0 V of d µ E be an intutionistic fuzzy graph. The a vertex ( ) i v d ν where where d be an intutionistic fuzzy graph. The iv is a pair of functions , where for all , where for all is a pair of functions is ( ) ∑ v defined = j i by ∈ , u v V ( ) ( ) v ) ( ) ( ) v µ and , µ i ij ≠ ( ) ∑ v = ν d ν ij ) ≠ i j be an intutionistic fuzzy graph, such Definition 2.5 The effective degree of a vertex v in intutionistic fuzzy graph, G = (V,E) is defined to be the sum of the fuzzy graph, G = (V,E) is defined to be the sum of the [ 1 , 0 ], µ → : V n v The effective degree of a vertex v in intutionistic 1 @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 2 | Issue – 6 | Sep-Oct 2018 Oct 2018 Page: 1473

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 ( ) G E δ and International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 3. Nil complementary in IFG Theorem 3.1 A dominating set S is a nil complementary dominating set if and only if it contains enclave. Proof Let S be a nil complementary dominating set of a intutionistic fuzzy graph G = dominating set which implies that there exit a vertex u∈ such that u µ ( v u v u 1 1 2 , min , ν ν ν < for u is an enclave of S. Hence S contain enclave. Conversely, suppose the dominating set S contains enclaves. Without loss of generality let us take the enclave of S. (i.e) µ ( v u v u 1 1 2 , min , ν ν ν < for all is not a dominating set. Hence dominating set S is nil complementary dominating set. Theorem 3.2 If S is a nil complementary dominating set of an intutionistic fuzzy graphG = vertex S u∈ such that { } S − Proof Let S be a nil complementary dominating set. since by theorem 3.1, every nil complementary dominating set if and only if it contains atleast one enclave. Let be an enclave of s. Then µ ( v u v u 1 1 2 , min , ν ν ν < for all connected intutionistic fuzzy graph, there exist atleast a vertex S w∈ such that µ ( v u v u 1 1 2 , min , ν ν ν = hence set. Theorem 3.3 A nil complementary dominating set in an intutionistic fuzzy graph G = Proof Let S be a nil complementary dominating set. Since by theorem 3.1, every nil complementary dominating set if and only if it contains atleast one enclave. Let S u ∈ be an enclave ( ) ( ) ( ) [ v u v u 1 1 2 , max , µ µ µ < Nil complementary in IFG strong edge incident at v. It is denoted by ( ) G E ∆ . The minimum ( ) v v d G E E = min δ The maximum ( ) v v d G E E = ∆ max Two vertices in intutionistic fuzzy graph if there is a strong edges between . Definition 2.6 An intutionistic fuzzy graph be an intutionistic fuzzy subgraph of V V ⊆ ′ and E E ⊆ ′ . That is µ µ ≤ ′ ; ij ij 2 2 γ γ ≤ ′ for every i,j=1,2,.....n Definition 2.7 Let ( E V G , = ,be an intutionistic fuzzy graph. The complement of an intutionistic fuzzy graph is, denoted by E V G , = , to be satisfied the following conditions. v V = (ii) and for all i=1,2,.....n ( ij j i ij 2 1 1 2 , min µ µ µ µ − = and for all i,j=1,2,...n Definition 2.8 Let ( E V G , = be an intutionistic fuzzy graph. A set V S ⊂ is said to be a nil complementary dominating set of an intutionistic fuzzy graph of G, if S is a dominating set and its complement V dominating set. The minimum scalar cardinality ove all nil complementary dominating set is called a nil complementary domination number and it is denoted by ncd γ , the corresponding complementary dominating set is denote by Definition 2.9 Let V S ⊂ in the connected intutionistic fuzzy graph ( E V G , = . A vertex S S if v u 2 max , µ < ( v u v u 1 1 2 , min , ν ν ν < ( ) S u N ⊆ . strong edge incident at v. It is denoted by A dominating set S is a nil complementary dominating set if and only if it contains at least one The minimum degree ) V . degree of of G G is is ( ( ) ∈ degree ) V degree of of G G is is Let S be a nil complementary dominating set of a ( ( ) ( ) ∈ V, E . The V-S is not a dominating set which implies that there exit a vertex [ ∈ ] ( ) v ( ) u V ( ) v . Therefore jv are said to be neighbourhood are said to be neighbourhood iv and < < µ µ S 2 , ( ) max , and S 1 1 [ ] ) ( ) in intutionistic fuzzy graph if there is a strong edges − v all jv iv and is an enclave of S. Hence S contain atleast one ′ ′ , ,E′ is said to ( V G = ; i 1 ν ν ≤ ′ = (V ) H Conversely, suppose the dominating set S contains enclaves. Without loss of generality let us take u be ) , E if [ − ] (u ) ( ) u . Hence V-S ( ) v < v µ µ 2 , , max V ∈ , v and 1 ′ µ ≤ µ ; and 1 S 1 1 1 i i i [ ] ) ( ) ( ) for all for every i,j=1,2,.....n 2 2 ij ij is not a dominating set. Hence dominating set S is nil complementary dominating set. ) ,be an intutionistic fuzzy graph. The complement of an intutionistic fuzzy graph is, denoted If S is a nil complementary dominating set of an ( ) ( ) following conditions. V, E , then there is a } is a dominating set. (i) u µ = µ ν = ν for all i=1,2,.....n 1 1 1 1 i i i i ( ν ) ) ν = ν − ν min min 1, (iii) 2 1 2 ij i j ij Let S be a nil complementary dominating set. since by theorem 3.1, every nil complementary dominating set if and only if it contains atleast one enclave. Let u∈ ] and S [ ( ( ) ∈ ( ) u ( ) v < µ . Since G is µ , max V − , u v v 2 1 1 ) be an intutionistic fuzzy graph. A set is said to be a nil complementary dominating set of an intutionistic fuzzy graph of G, if S is a dominating set and its complement V-S is not a dominating set. The minimum scalar cardinality over all nil complementary dominating set is called a nil complementary domination number and it is denoted , the corresponding complementary dominating set is denote by [ ] ) ( ) ( ) all S connected intutionistic fuzzy graph, there exist atleast [ ] (u u ) ( ) u ( ) v and = µ µ , max { } u , v S − 2 1 1 [ ] ) ( ) ( ) is a dominating minimum minimum nil nil A nil complementary dominating set in an -set. γ ( ( ) V, E is not a singleton. ncd in the connected intutionistic fuzzy graph is said to be an enclave of Let S be a nil complementary dominating set. Since by theorem 3.1, every nil complementary dominating set if and only if it contains atleast one enclave. Let be an enclave ) u∈ is said to be an enclave of ) u 1 , µ µ ( ) for [ ( )] v V− ( ( ) allv∈ and (i.e) 1 1 ∈ [ ] ) ( ) of of s. s. Then Then and S . ] @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 2 | Issue – 6 | Sep-Oct 2018 Oct 2018 Page: 1474

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 ) ( ) ( ) [ ] v u v 1 1 , min ν ν < for all V v − ∈ contains only one vertex u , then it must be isolated in G. This is contradiction to connectedness. Hence nil complementary dominating set contains more than one vertex. Conclusion The nil complementary dominating set and its number in intutionistic fuzzy graph is defined. Some of the properties are derived in this paper. The future work will be carried on its application in social network. References 1.Atanassov. K. T. Intutionistic fuzzy se and its Applications. Physica, New York, 1999. 2.Bhattacharya, P. Some remarks on fuzzy graphs, Pattern Recognition Letter 5: 297-302 3.Harary, F., Graph Theory, Addition Wesley, Third Printing, October 1972. 4.C. V. R. Harinarayanan and P. Muthuraj and P. J. Jayalakshmi, “Total strong (weak) domination intutionistic fuzzy graph”, Advances in theoretical and applied mathematics, 11 (2016), 203 , 11 (2016), 203-212. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 ( Advances in soft computing: Computatinal intelligence; Theory and Applications, Verlag, 20 (2006), 139-150 6.M. G. Karunambigai and R. Parvathi and R. Buvaneswari, “Constant graphs”, Notes on Intuitionistic Fuzzy system, (2011), 1, 37-47. 7.M. G. Karunambigai and R. Parvathi and R. Buvaneswari, “Arcs in intutionistic fuzzy graph”, Notes on Intuitionistic Fuzzy system 8.M. G. Karunambigai and Muhammad Akram and R. Buvaneswari, “Strong and super strong vertices in intutionistic fuzzy graphs”, intuitionistic fuzzy system, 9.Somasundaram, A and Somasundaram, S., Domination in Fuzzy graph Letter 19(9), 1998, 787-791. 10.Tamizh Chelvam. T and Robinson Chelladuari. S, Complementary nil domination number of a graph, Tamkang Journal of Mathematics, No.2 (2009), 165-172. Received: June 11.Nagoor Gani. A and Shajitha Begum. Order and Size in Intuitionistic Fuzzy Graphs, International Journal of Algorithms, Computing and Mathematics, (3) 3 (2010). (3) 3 (2010). Advances in soft computing: Computatinal intelligence; Theory and Applications, Springer- 150 . Suppose S ν − , u S 2 , then it must be isolated in G. This is contradiction to connectedness. Hence nil complementary dominating set contains more than M. G. Karunambigai and R. Parvathi and R. Buvaneswari, “Constant n Intuitionistic Fuzzy system, 17 intutionistic intutionistic fuzzy fuzzy complementary dominating set and its number in intutionistic fuzzy graph is defined. Some of the properties are derived in this paper. The future work will be carried on its application in social network. M. G. Karunambigai and R. Parvathi and R. Buvaneswari, “Arcs in intutionistic fuzzy graph”, Notes on Intuitionistic Fuzzy system, 17 (2011). M. G. Karunambigai and Muhammad Akram and R. Buvaneswari, “Strong and super strong vertices in intutionistic fuzzy graphs”, Journals on T. Intutionistic fuzzy sets: Theory and its Applications. Physica, New York, 1999. 30 (2016), 671-678. Bhattacharya, P. Some remarks on fuzzy graphs, Somasundaram, A and Somasundaram, S., Domination in Fuzzy graph-I, Patter Recognition 791. 302, 1987. Harary, F., Graph Theory, Addition Wesley, Third Tamizh Chelvam. T and Robinson Chelladuari. S, Complementary nil domination number of a Journal of Mathematics, Vol.40, Received: June huraj and P. J. Jayalakshmi, “Total strong (weak) domination Advances in theoretical Nagoor Gani. A and Shajitha Begum. S, Degree, Order and Size in Intuitionistic Fuzzy Graphs, International Journal of Algorithms, Computing 5.M. M. “Intuitionisitc Fuzzy Graph”, Proceedings of 9th International Conference Intelligence, G. G. Karunambigai Karunambigai and and R. R. Parvathi, Parvathi, “Intuitionisitc Fuzzy Graph”, Proceedings of 9 Fuzzy Days International Conference Intelligence, @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 2 | Issue – 6 | Sep-Oct 2018 Oct 2018 Page: 1475