BONDAGE AND NON-BONDAGE NUMBER OF A FUZZY GRAPH

: In this paper, bondage and non-bondage set of a fuzzy graph are discussed. The bondage number b ( G ) and non-bondage number b n ( G ) of a fuzzy graph G are deﬁned. The upper bound for both b ( G ) and b n ( G ) are given. Also some results on b ( G ) and b n ( G ) are discussed. The exact values of b ( G ) and b n ( G ) are determined for several classes of fuzzy graphs.


Introduction
Cockayne and Hedetniemi [2] introduced the domination number and the independent domination number of graphs but the concept of dominating sets in graphs was introduced by Ore and Berge [1,10].In 1990, the concept of the bondage number in graphs was introduced by Fink, Jacobson, Kinch and Roberts [3].Later in 1994, Hartnell and Douglas.F. Rall [4] discussed about the bounds on the bondage number.Kulli and Janakiram [5] introduced the non-bondage number in graphs.The concept of fuzzy relation was introduced by Zadeh [14] in his classical paper in 1965.Rosenfeld [11] introduced the notion of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as paths, cycles and connectedness.In 1998, A. Somasundram and S. Somasundram [12] discussed domination in fuzzy graphs using effective edges.Nagoor Gani and Chandrasekaran [6] discussed domination in fuzzy graph using strong arcs.Nagoor Gani and Vadivel [9] discussed domination, independent domination and irredundance in fuzzy graphs using strong arcs.Nagoor Gani and Prasanna Devi [7] discussed edge dominating set and fuzzy edge independent set in fuzzy graphs.

Preliminaries
The bondage number of a graph G is the minimum cardinality of a set of edges of G whose removal from G results in a graph with domination number larger than that of G.The non-bondage number of a graph is the maximum cardinality among all sets of edges X ⊆ E such that the domination number of G−X is same as the domination number of G.
A fuzzy graph G =< σ, µ > is a pair of functions σ : V → [0, 1] and µ : V × V → [0, 1], where for all x, y ∈ V , we have µ(x, y) ≤ σ(x)∧σ(y).The underlying crisp graph of a fuzzy graph G =< σ, µ > is denoted by G * =< σ * , µ * >, where there exist u ∈ D such that u dominates v.The domination number,γ(G), is the smallest number of nodes in any dominating set of G.A subset D of E(G) is said to be an edge dominating set of G if for every e j ∈ E(G) − D there exist e i ∈ D such that e i dominates e j .The smallest number of edges in any edge dominating set of G is called its edge domination number and it is denoted by γ ′ (G).

Bondage Number
In this section we define bondage set and bondage number of fuzzy graphs.The upper bound for the bondage number is also discussed and some results are given.
Definition 3.1.Let G be a fuzzy graph.If there exist a set X ⊆ S such that γ(G − X) > γ(G) then X is said to be a bondage set of G, where S is the set of all strong arcs in G.    Suppose e does not becomes a strong arc in If n = 3m + 1 then γ(G) = m + 1.The domination number increases only if we delete minimum 3 strong arcs.Therefore b(G) = 3.
If n = 3m + 1 then the domination number increases when we delete minimum 2 strong arcs adjacent to the same node.Therefore b(G) = 2.   • {e 1 , e 2 } is a non-bondage set of the fuzzy graph G.This set is not an edge dominating set of G because the edge e 5 is not dominated by any edge in {e 1 , e 2 }.
• {e 3 , e 4 } is also a non-bondage set of G.But this set is an edge dominating set of G because each edge e 1 , e 2 and e 5 are dominated by the set {e 3 , e 4 }.
Thus the non-bondage set of a fuzzy graph G need not be an edge dominating set of G.  Proof.Let G be a fuzzy graph and it does not have a bondage set.i.e., there does not exist any set X ⊆ S such that γ(G − X) > γ(G).Thus deletion of all strong arcs from G does not increases the domination number of G. Now delete the set of all strong arcs, S, and the domination number will be γ(G−S) = γ(G).Therefore b n (G) = |S|.
Theorem 5.12.If G is a complete fuzzy graph with n vertices or nodes then Proof.Let G be a complete fuzzy graph with n vertices.In G, all arcs are strong arcs.Thus the total number of (strong) arcs in G are n(n − 1)/2.
We know that γ(G) = 1.Each node will dominate all other nodes.Therefore we need minimum Proof.Let G be a fuzzy graph and G* be a star.Then the domination number of G is 1 i.e., γ(G) = 1.Thus the node in the centre of G dominates all other nodes in G.
So deletion of any one arc of G will result γ(G) = 2 since all arcs of G are strong arcs in G. Thus we don ′ t have a non-bondage set for G. Therefore b n (G) = 0.

Conclusion
We discussed about the bondage number of fuzzy graphs and its upper bound.We also given bondage number of a complete fuzzy graph.The non-bondage number of a fuzzy graph is also defined.The exact value of the non-bondage number of the fuzzy graphs and the relation between bondage number and non-bondage number of the fuzzy graphs are given.Using these concepts some future work are to find the bondage number and the non-bondage number for fuzzy trees and also to find the lower bound of both bondage and non-bondage number.

Definition 5 . 1 .
Case (ii) Suppose G has only one weakest arc, say e, then G has n-1 strong arcs and deletion of any one strong arc makes the weakest arc as a strong arc in G − e 1 , e 1 ( = e) ∈ S. Clearly b(G) = 3 if n = 3m + 1 and b(G) = 2 if n = 3m + 1 but the weakest arc does not belongs to any bondage set of G. Corollary 4.4.Let P n be a fuzzy graph and it is a path with n(≥ 2)nodes then b(P n ) = 2, if n = 3m + 1, m = 1, 2, .... 1,otherwise5.Non-Bondage NumberIn this section, non-bondage set and non-bondage number of a fuzzy graph are defined.Some results on non-bondage number are given.The upper bound for non-bondage number is also defined.And non-bondage number for complete fuzzy graph is given.The set of strong arcs X ⊆ S is called a non-bondage set if γ(G − X) = γ(G) where S is the set of all strong arcs in G.Example 5.2.

Theorem 5 . 9 .
If a fuzzy graph G does not have a bondage set then b n (G) = |S|.

Theorem 5 . 10 .Theorem 5 . 11 .
If a fuzzy graph G has a bondage set then b(G) ≤ b n (G)+1.Proof.Let G be a fuzzy graph which has a bondage set.A b n -set is a maximum non-bondage set i.e., deletion of all arcs in a b nset results in γ(G) = γ(G − b n ).So deletion of any strong arc e / ∈ b n with the edges in the set b n results in γ(G − {b n ∪ e}) > γ(G) which implies {b n ∪ e} is a bondage set.Thus b(G) ≤ |b n ∪ e| = b n (G) + 1 =⇒ b(G) ≤ b n (G) + 1.If a non-bondage set of G is an edge dominating set of G then b n (G) ≥ γ(G)/2.Proof.Let G be a fuzzy graph.Let D be a non-bondage set of G and it is also an edge dominating set of G. Clearly |D| ≥ γ ′ (G) and |D| ≤ b n n − 1 arcs to keep γ(G) = 1.Thus we can almost delete |S| − (n − 1) arcs.Therefore bn (G) =|S| − (n − 1) =n(n − 1)/2 − (n − 1) =(n − 1)(n/2 − 1) =(n − 1)((n − 2)/2), b n (G) =(n − 1)(n − 2)/2.Theorem 5.13.If G is a fuzzy graph and G* is a star then b n (G) = 0.