SOME RESULTS ON CONNECTED ACCURATE EDGE DOMINATION IN FUZZY GRAPHS USING STRONG ARCS

I n this paper, connected accurate edge dominating set of a fuzzy graph is discussed. An accurate edge dominating set D is said to be a connected accurate edge dominating set, if an induced fuzzy subgraph <D> of G is connected. The connected accurate edge domination number, 𝛾 𝑓𝑐𝑎𝑒 ( 𝐺 ) is the minimum fuzzy cardinality taken over all connected accurate edge dominating sets of G. We also determine, upper bounds of connected accurate edge domination number for some standard fuzzy graphs. Relations between connected accurate edge domination number and some other edge domination parameters are discussed.


INTRODUCTION
Ore [11] and Berge [1] introduced the concept of dominating sets in graphs. Then, the domination number and independent domination number was introduced by Cockayne and Hedetniemi [2]. An accurate domination and connected accurate domination was introduced by Kulli and Kattimani [3,4]. And also, accurate edge domination number in graph was introduced by Kulli and Kattimani [5]. Kulli and Srgarkanti [6] introduced the concept of connected edge domination number of a graph. Then accurate connected edge domination number in graphs was studied by Venkanagouda M. Goudar et al. [14].
The strong neighbourhood edge degree of an edge is, . The minimum strong neighbourhood edge degree of a fuzzy graph G is defined as, The maximum strong neighbourhood edge degree of a fuzzy graph G is defined as, A fuzzy graph = 〈 , 〉 is said to be a complete fuzzy graph, if � , � = ( ) ⋀ � � for all , ∈ * .
A fuzzy graph = 〈 , 〉 is said to be a connected fuzzy graph, if there exists a strong path between every pair of nodes.
A subset ′ of S is said to be an edge dominating set of G, where ⊆ ( ), set of all strong arcs in ( ), if for every ∈ ( ) − ′ there exists ∈ ′ such that dominates . The minimum cardinality taken overall edge dominating sets of fuzzy graph G is called an edge domination number and it is denoted by ( ).
Let a subset ′ of S be an edge dominating set of fuzzy graph G and it is said to be an accurate edge dominating set of G, if − ′ has no edge dominating set with same fuzzy cardinality | ′|. The accurate edge domination number, which is denoted as ( ), of a fuzzy graph G is the minimum fuzzy cardinality taken over all accurate edge dominating sets of G.

CONNECTED ACCURATE EDGE DOMINATING SET IN FUZZY GRAPH
Definition 3.1: Let be the connected fuzzy graph and ′ ⊆ , where ⊆ ( ), set of all strong arcs in ( ).Then the accurate edge dominating set ′ is said to be a connected accurate edge dominating set of fuzzy graph G, if the induced fuzzy graph 〈 ′〉 is a connected fuzzy graph.
The connected accurate edge domination number of fuzzy graph G, is the minimum fuzzy cardinality taken over all connected accurate edge dominating sets of G, and it is denoted by γ ( ). From the above fig. 3.1, some of the edge dominating sets are 1

Remark:
(i) The connected accurate edge dominating set of a fuzzy graph G may or may not be a minimal edge dominating set. (ii) Every minimum connected accurate edge dominating set of a fuzzy graph G is an accurate edge dominating set and also edge dominating set of G. Let a subset ′ of S, be the minimum edge dominating set of a fuzzy graph G, then ( ) = | ′ |. Suppose that, 〈 − ′〉 has no edge dominating set with same fuzzy cardinality | ′ |, then D' itself forms an accurate edge dominating set of G. Therefore, ( ) = ( ).
Then, from the result, every minimum accurate edge dominating set, 1 ′ ⊆ , of a fuzzy graph G is also the edge dominating set of G. Then, the accurate edge domination number is Similarly, by case (i), Let ′ be the minimum accurate edge dominating set of fuzzy graph G. If 〈 ′〉 is connected, then ′ be the connected accurate edge dominating set of G and the connected accurate edge domination number of fuzzy graph G, is ( ) = | ′|.
Also, every minimum connected accurate edge dominating set of a fuzzy graph G is an accurate edge dominating set of G.
That is, From (3.1) and (3.2), we get, Since, Proof: Let G be a connected fuzzy graph. Let a subset ′ of S, ⊆ ( ) be the set of all strong arcs, be the edge dominating set of G. If 〈 ′〉 is connected, then ′ be the connected edge dominating set of G and the connected edge domination number of fuzzy graph G, is ( ) = | ′| and | ′| be the minimum fuzzy cardinality taken over all edge dominating sets of G.
Similarly, If 〈 − ′〉 does not contain any edge dominating set of same fuzzy cardinality | ′| then, ′ itself forms an accurate edge dominating set of G. If, the accurate edge dominating set, ′, of the fuzzy graph G is with minimum fuzzy cardinality among all accurate edge dominating set of G and the fuzzy subgraph induced by 〈 ′〉 is connected, then, the accurate edge dominating set ′ itself forms the connected accurate edge dominating set of G.