INVERSE DOMINATION ON MULTI FUZZY GRAPH

In this paper, inverse domination on multi fuzzy graph, inverse total domination of a multi fuzzy graph are introduced. The properties of inverse domination number and inverse total domination number on multi fuzzy graph are discussed. The necessary and sufficient condition for the existence of an inverse dominating set and an inverse total dominating set are established. Some basic theorems related to the stated dominations have also been presented.


INTRODUCTION
The study of domination in graphs was established by O. Ore [13] and C. Berge [1].
T.W. Haynes, S.T. Hedetniemi and P.J. Slater [5] was introduced the Fundamentals of domination 5253 INVERSE DOMINATION ON MULTI FUZZY GRAPH in graphs. E.J. Cockayne and S.T. Hedetniemi [2] initiated Total domination in graphs. The inverse domination in graphs concept proposed by V.R. Kulli and S.C. Sigarkanti [8]. V.R. Kulli and R.R. Iyer [7] discussed the concept of Inverse total domination in graphs. The concept of fuzzy graph was first introduced by Kaufmann [6] from the concept fuzzy relation introduced by L.A. Zadeh [17]. A. Nagoorgani and V.T. Chandrasekaran [12] was introduced the concept of domination in fuzzy graph using strong arcs. C.V.R. Hari Narayanan, S. Revathi and R. Muthuraj [4] was discussed connected total perfect dominating set in fuzzy graph. S. Ghobadi, N.D. Soner and Q.M. Mahyoub [3] was discussed inverse dominating set in fuzzy graph. A. Somasundaram and S. Somasundaram [15,16] discussed about domination in fuzzy graph with effective edges and also discussed Total domination in fuzzy graph using effective edges. R. Muthuraj et al. [10,9] was discussed On Anti fuzzy graph and Total Strong (Weak) Domination on Anti fuzzy graph. S. Sabu and T.V. Ramakrishnan [14] proposed the theory of multi-fuzzy sets and it is used to characterize the problems in the field of taste recognition, decision making, pattern recognition, image processing and approximation of vague data. R. Muthuraj and S. Revathi [11] was defined the concept of Multi fuzzy graph. In this paper we obtain the concept of Inverse domination on Multi fuzzy graph and Inverse total domination of a Multi fuzzy graph. We defined some properties and the necessary and sufficient condition for the inverse and inverse total domination number of a Multi fuzzy graph.

PRELIMINARIES: MULTI FUZZY GRAPH
In this section, the concept of multi fuzzy graph is introduced and discussed its related concepts. Throughout this paper, denote the edge between two vertices u and v as uv.

Remark:
i. If the sequences of the membership functions have only k-terms (finite number of terms), k is called the dimension of A.

Definition [11]
A multi fuzzy graph (MFG) of dimension m defined on the underlying crisp graph

Definition [11]
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Then the vertex cardinality of a MFG G or the order of a MFG is denoted as V or O(G) or p and is defined as

Definition [11]
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Let D  V. Then the cardinality of D of G or the fuzzy cardinality of D of G is denoted as D and is defined as

Definition [11]
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Let u  V of G. Then the neighbors (neighborhood) of u or an open neighbors of u V of G is denoted by N(u) and is The closed neighbors of u  V of G is denoted by N[u] and is defined as
In other words, the vertex u  V of G is said to be isolated vertex if N(u) =  or  N(u)  = 0.

Definition [11]
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. The vertex u  V of G is said to be pendant vertex if  N(u)  = 1. An edge incident on the pendant vertex is called a pendant edge.

Definition [11]
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Let u, v  D. We say The minimum fuzzy cardinality of a minimal dominating set in G is called the domination number of G and is denoted by (G) or simply  and the corresponding minimal dominating set is called the minimum dominating set of G.

Definition [11]
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Then a dominating set T of V is said to be a total dominating set T of G if the induced sub graph of T has no isolated vertex.
A total dominating set T of a MFG G is called minimal total dominating set on MFG G, if no proper subset of T is a total dominating set of G.
The minimum fuzzy cardinality among all minimal total dominating set on MFG G is called total dominating number of a MFG G and is denoted by T γ (G) and the corresponding minimal dominating set is called is called the minimum dominating set of G.

Example
In the following MFG with dimension 3,

MAIN RESULTS: INVERSE DOMINATION ON MFG
In this section, the concept of inverse domination on MFG is defined and its domination number is obtained for MFG with examples. Throughout this section, domination on MFG using effective edges only considered.

Definition
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Let I be a minimum dominating set of a MFG G. If the subset I V -I is a dominating set of G then I is called an inverse dominating set with respect to I.
The minimum fuzzy cardinality taken over all minimal inverse dominating sets in G is called inverse domination number of G and it is denoted by .

Example
Consider the following MFG G.

Remark
The following theorem is the necessary and sufficient conditions for the existence of at least one inverse dominating set of a MFG G.

Theorem
Let

MAIN RESULTS: RELATED CONCEPTS OF INVERSE TOTAL DOMINATION ON MFG
In this section, the related concepts of inverse total domination on MFG and its properties are discussed.

Definition
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Let T be a minimum total dominating set of a MFG G. If the subset TI  V -T is a total dominating set of G, then TI is called an inverse total dominating set with respect to T.
The minimum fuzzy cardinality taken over all minimal inverse total dominating set in G is called inverse total domination number of G and is denoted by (G). γ T I 5260 R. MUTHURAJ, S. REVATHI

Example
Consider a MFG G with dimension 3.

Remark
The following theorem is the necessary and sufficient conditions for the existence of at least one inverse total dominating set of a MFG G.

Theorem
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m. Let T be a minimal total T is a total dominating set of V in G then, by the definition 2.23, n(T) 2.
ii. TI is an inverse total dominating set of G then n(G)  4.

Remark
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a MFG of dimension m with n vertices and n edges, where n = 4, 8, 12, …. Let T be a minimum total dominating set of G. Then, an inverse total dominating set TI exists with respect to T and .

Theorem
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a complete MFG of dimension m. Let TI be an inverse total dominating set with respect to the minimum total dominating set T of G. Then both  T  and  TI are connected.

Proof
Let G = ((1, 2, ….,m), (1, 2 , …., m)) be a complete MFG of dimension m. Let TI be an inverse total dominating set with respect to the minimum total dominating set T of G.
Let uT, then u is adjacent to all vertices of V in G and also adjacent to all vertices of T.
That is, there is a fuzzy path between all vertices of T in G. Hence,  T  is connected.
Let vTI, then v is adjacent to all vertices of V in G and also adjacent to all vertices of TI.
That is, there is a fuzzy path between all vertices of TI in G. Hence,  TI is connected.
Then, a minimum total dominating set T and an inverse total dominating set TI of G exists.