Complementary Nil Domination in Intuitionistic Fuzzy Graph

In this paper, We define complementary nil dominating set and its numbers of Intuitionistic Fuzzy Graph. The bound on this number are obtained for some standard Intuitionistic fuzzy graphs are derived. Some theorems related to the above concepts are studied. Mathematics Subject Classification: 05C69, 03F55, 05C72, 03E72


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.Complementary nil domination set in the crisp graph introduced and analyzed by Tamizh Chelvam et.al., [6] in 2009.
In this paper, We define complementary nil dominating set and its numbers of Intuitionistic Fuzzy Graphs.The bound on this number are obtained for some standard Intuitionistic fuzzy graphs.Some theorems related to the above concepts are studied and this concept is useful in networking analysis.

Definition 2.11:
The neighbourhood degree of a vertex is defined as dN and dNγ(v . The minimum neighbourhood degree is defined as N(G) = (Nµ(v), Nγ(v)), where Nµ(v) = Λ { dNµ(v): v∈V} and Nγ(v) = Λ { dNγ(v): v∈V}.Definition 2.12: 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 Є V} Definition 2.13: 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 subset D ⊆V is said to be dominating

R. Jahir Hussain and S. Yahya Mohamed
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 dominating set is called domination number and is denoted by γ.The maximum scalar cardinality of a minimal domination set is called upper domination number and is denoted by the symbol Γ .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.

Complementary nil domination set in IFG
Definition 3.1: Let G = (V, E) be an IFG.A set S ∁ V is said to be a complementary nil domination set (or simply cnd-set) of an IFG G, if S is a dominating set and its complement V-S is not a dominating set.The minimum scalar cardinality over all cnd-set is called a complimentary nil domination number and is denoted by the symbol γcnd, the corresponding minimum cnd -set is denote by γcnd-set.In the above Fig- 1, v1 is enclave of the set S1 and v3 is enclave of the set S2.

Theorem 3.4:
A dominating set S is a cnd-set if and only if it contains at least one enclave.