SET DOMINATION IN FUZZY GRAPHS USING STRONG ARCS

A BSTRACT . Set domination in fuzzy graphs is very useful for solving trafﬁc problems during communication in computer networks and travelling networks. In this article, the concept of set domination in fuzzy graphs using strong arcs is introduced. The strong set domination number of complete fuzzy graph and complete bipartite fuzzy graph is determined. It is obtained the properties of the new parameter and related it to some other known domination parameters of fuzzy graphs. An upper bound for the strong set domination number of fuzzy graphs is also obtained.


INTRODUCTION
Fuzzy graphs were introduced by Rosenfeld, who has described the fuzzy analogue of several graph theoretic concepts like paths, cycles, trees and connectedness [22]. Bhutani and Rosenfeld have introduced the concept of strong arcs [7]. The work on fuzzy graphs was also done by Mordeson, Pradip, Talebi, and Yeh [16,21,32,33]. It was during 1850s, a study of dominating sets in graphs started purely as a problem in the game of chess. Chess enthusiasts in Europe considered the problem of determining the minimum number of queens that can be placed on a chess board so that all the squares are either attacked by a queen or occupied by a queen. The concept of domination in graphs was introduced by Ore and Berge in 1962, the domination number and independent domination number are introduced by Cockayne and Hedetniemi [11]. Connected domination in graphs was discussed by Sampathkumar and Walikar [23]. Somasundaram and Somasundaram discussed domination in fuzzy graphs. They defined domination using effective edges in fuzzy graph [26,27]. Nagoorgani and Chandrasekharan defined domination in fuzzy graphs using strong arcs [18]. Manjusha and Sunitha discussed some concepts in domination and total domination in fuzzy graphs using strong arcs [12,13]. In this paper it is discussed set domination in fuzzy graphs using strong arcs.

PRELIMINARIES
It is quite known that graphs are simply models of relations. A graph is a convenient way of representing information involving relationship between objects. The objects are represented by vertices and relations, by edges. When there is vagueness in the description of the objects or in its relationships or in both, it is natural that we need to design a 'fuzzy graph model'. We summarize briefly some basic definitions in fuzzy graphs which are presented in [6, 7, 12, 17-19, 22, 26, 28].
A node u is said to be isolated if µ(u, v) = 0 for all v = u.

STRONG SET DOMINATION IN FUZZY GRAPHS
The concept of domination in graphs was introduced by Ore and Berge in 1962 ,the domination number and independent domination number are introduced by Cockayne and Hedetniemi [11]. Set domination in graphs was discussed by Sampathkumar and Pushpalatha [24]. For the terminology of domination and set domination in crisp graphs we refer to [10,24].
For a node v of a graph G : (V, E), recall that a neighbor of v is a node adjacent to v in G. Also A node v in a graph G is said to dominate itself and each of its neighbors, that is v dominates the nodes in N [v]. A set S of nodes of G is a dominating set of G if every node of V (G) − S is adjacent to some node in S. A minimum dominating set in a graph G is a dominating set of minimum cardinality. The cardinality of a minimum dominating set is called the domination number of G and is denoted by γ(G). A dominating set S ⊂ V is a set dominating set of G if for every set D ⊂ V − S, there exists a a non empty subset E ⊂ S such that the sub graph < D E > is connected. The minimum of the cardinalities of the set dominating sets of G is termed as the set domination number of G, and is denoted as γ s (G).
Nagoorgani and Chandrasekaran [18] introduced the concept of domination using strong arcs in fuzzy graphs. According to Nagoorgani a node v in a fuzzy graph G is said to strongly dominate itself and each of its strong neighbors, i.e., v strongly dominates the nodes in N s [v]. A set D of nodes of G is a strong dominating set of G if every node of V (G) − D is a strong neighbor of some node in D. They defined a minimum strong dominating set in a fuzzy graph G as a strong dominating set with minimum number of nodes [18]. An equivalent definition which provides much motivation for this paper as follows A set D ⊂ V is a strong dominating set of G if for every singleton subset {u} of V − D, there exists a singleton subset {v} of D such that < {u, v} > is connected.
Manjusha and Sunitha [14] defined strong domination number using membership values (weights) of arcs in fuzzy graphs as follows. Definition 1.1. [14] The weight of a strong dominating set D is defined as W (D) = u∈D µ(u, v), where µ(u, v) is the minimum of the membership values(weights) of strong arcs incident on u. The strong domination number of a fuzzy graph G is defined as the minimum weight of strong dominating sets of G and it is denoted by γ s (G) or simply γ s . A minimum strong dominating set in a fuzzy graph G is a strong dominating set of minimum weight.
Let γ s (G) or γ s denote the strong domination number of the complement of a fuzzy graph G. Now the set domination in fuzzy graphs using strong arcs is defined as follows.
is the minimum of the membership values(weights) of strong arcs incident on u. The strong set domination number of a fuzzy graph G is defined as the minimum weight of strong set dominating sets of G and it is denoted by γ ss (G) or simply γ ss . A minimum strong set dominating set in a fuzzy graph G is a strong set dominating set of minimum weight.
Let γ ss (G) or γ ss denote the strong set domination number of the complement of a fuzzy graph G.
Apart from obtaining bounds for γ ss (G), we relate to some other known parameters of G. Throughout this article, by a fuzzy graph we mean a connected fuzzy graph. A strong dominating set D of a fuzzy graph G : (V, σ, µ) is a strong connected dominating set of G if the induced fuzzy subgraph < D > is connected. The weight of a strong connected dominating set D is defined as W (D) = u∈D µ(u, v), where µ(u, v) is the minimum of the membership values(weights) of strong arcs incident on u. The strong connected domination number of a fuzzy graph G is defined as the minimum weight of strong connected dominating sets of G and it is denoted by γ sc (G) or simply γ sc . A minimum strong connected dominating set in a fuzzy graph G is a strong connected dominating set of minimum weight.  Proof. Let D be a strong set dominating set of H. Since H is a maximum spanning tree of G we have σ = ν. Hence the nodes in D strongly set dominates all the nodes in V \ D. Hence D is a strong set dominating set of G. Hence γ ss (G) ≤ γ ss (H).

STRONG SET DOMINATION IN CLASSES OF FUZZY GRAPHS
In this section, it is determined the strong set domination number of complete fuzzy graph, complete bipartite fuzzy graph, fuzzy cycles and join of a fuzzy graph with a complete fuzzy graph.
Proof: Since G is a complete fuzzy graph, all arcs are strong [29] and each node is adjacent to all other nodes. Hence D = {u} is a strong set dominating set for each u ∈ σ * . Hence the result follows.
where µ(u, v) is the weight of a weakest arc in K σ1,σ2 .
Proof: In K σ1,σ2 , all arcs are strong. Also each node in V 1 is adjacent with all nodes in V 2 . Hence in K σ1,σ2 , the strong set dominating sets are V 1 , V 2 and any set containing at least 2 nodes, one in V 1 and other in V 2 . Among this if V 1 or V 2 contains only one element say u, then D = {u} is the minimum strong set dominating set in G. Hence γ ss (K σ1,σ2 ) = µ(u, v) where µ(u, v) is the minimum weight of arcs incident on u. If both V 1 and V 2 contains more than one element then the set {u, v} of nodes of any weakest arc (u, v) in K σ1,σ2 forms a strong set dominating set. Hence γ ss (K σ1,σ2 ) = µ(u, v) + µ(u, v) = 2µ(u, v). Hence the result. Proof: In a fuzzy cycle every arc is strong. Also, the number of nodes in a strong set dominating set of both G and G * are same because each arc in both graphs are strong. In graph G * , the strong set domination number of G * is obtained as (n − 3) [24]. Hence the minimum number of nodes in a strong set dominating set of G is (n − 3). Hence the result follows.
Proof: For any fuzzy graph G, any node in K σ is adjacent to all other nodes in K σ and G and note that all such arcs are strong arcs. Hence any singleton set D = {u} for each node u in K σ , is a strong set dominating set of K σ + G. Hence γ ss (K σ + G) = µ(u, v) where µ(u, v) is the weight of a weakest arc incident on u for any node u ∈ K σ .

MINIMAL STRONG SET DOMINATION IN FUZZY GRAPHS
In this section it is defined minimal strong set dominating sets and discussed some properties.    Note that in a complete fuzzy graph the minimum and minimal strong set dominated sets are same and any singleton set of nodes is the minimum strong set dominating set. Hence the following theorems are obvious.
Theorem 3.4. Every non trivial complete fuzzy graph G has a strong set dominating set D whose complement V \ D is also a strong set dominating set. Theorem 3.5. Let G be a complete fuzzy graph. If D is a minimal strong set dominating set then V \ D is a strong set dominating set.
Note that in a complete bipartite fuzzy graph the end nodes of any weakest arc forms a minimal strong set dominating set. Hence the following theorems are obvious. Theorem 3.6. Every non trivial complete bipartite fuzzy graph G has a strong set dominating set D of two elements whose complement V \ D is also a strong set dominating set. Theorem 3.7. Let G be a complete bipartite fuzzy graph. If D is a minimal strong set dominating set of two elements then V \ D is a strong set dominating set.
Remark 3.8. Theorems 5.4 to 5.7 are not true in general connected fuzzy graphs as seen in the following example. Example 3.9. Consider the fuzzy graph given in Figure 5.

STRONG SET DOMINATION IN FUZZY TREES
Note that in the definition of a fuzzy tree, F is the unique maximum spanning tree (MST) of G [31]. An arc is called a fuzzy bridge of a fuzzy graph G : (V, σ, µ) if its removal reduces the strength of connectedness between some pair of nodes in G [22]. Similarly a fuzzy cut node w is a node in G whose removal from G reduces the strength of connectedness between some pair of nodes other than w [22]. A node z is called a fuzzy end node if it has exactly one strong neighbor in G [8]. A non trivial fuzzy tree G contains at least two fuzzy end nodes and every node in G is either a fuzzy cut node or a fuzzy end node [8]. In a fuzzy tree G an arc is strong if and only if it is an arc of F where F is the associated unique maximum spanning tree of G [7,31]. Note that these strong arcs are α-strong and there are no β-strong arcs in a fuzzy tree [28]. Also note that in a fuzzy tree G an arc (x, y) is α-strong if and only if (x, y) is a fuzzy bridge of G [28]. Proof: Let D be a strong set dominating set of G. Let u ∈ D. Since D is a strong dominating set, there exists v ∈ V \ D such that (u, v) is a strong arc. Then (u, v) is an arc of the unique MST F of G [7,31]. Hence (u, v) is an α−strong arc or a fuzzy bridge of G [22]. Since u is arbitrary, this is true for every node of the strong set dominating set of G. This completes the proof.
Proposition 4.2. In a non trivial fuzzy tree G : (V, σ, µ), no node of a strong set dominating set is an end node of a β−strong arc.
Proof: Note that a fuzzy graph is a fuzzy tree if and only if it has no β−strong arcs [28]. Hence the proposition.

Theorem 4.3.
In a non trivial fuzzy tree G : (V, σ, µ), except K 2 , the set of all fuzzy cut nodes is a strong set dominating set.
Proof: Let D be the set of all fuzzy cut nodes of a non trivial fuzzy tree G : (V, σ, µ). Then D is a strong dominating set in G [12] and induced fuzzy subgraph < D > is connected [15]. Note that the strong neighbour of a fuzzy end node is a fuzzy cut node Hence for every proper subset S of V − D there exists a nonempty proper subset T of D such that S ∪ T is connected. Hence D is a strong set dominating set of G.
Remark 4.4. The set of all fuzzy end nodes need not be a strong set dominating set in a non trivial fuzzy tree G : (V, σ, µ) except K 2 .
Theorem 4.5. In a fuzzy tree G : (V, σ, µ), each node of every strong set dominating set is contained in the unique maximum spanning tree of G.
Proof: Since G is a fuzzy tree, G has a unique maximum spanning tree F which contains all the nodes of G. In particular, F contains all nodes of every strong set dominating set of G. This completes the proof. In case 1 it is obvious that W (S ) > W (S).
In case 2 < S > (the fuzzy sub graph induced by S ) is not connected if w is an internal node of < S > (the fuzzy sub graph induced by S) or S is not a strong dominating set if w is an end node of the fuzzy subgraph < S > for, A fuzzy tree contains at least 2 fuzzy end nodes. If w is an end node of < S > then one neighboring node of w is a fuzzy end node say u in G and w is the only strong neighbor of u in G. Therefore, if w is not contained in < S > then u is not strongly dominated by any node in G. Hence S is not a strong dominating set of G. a]. G has a unique maximum weighted arc adjacent to any fuzzy end node, then W (S ) > W (S) since weight of maximum arc is contributed to W (S ) but not to W (S) b]. The unique maximum weighted arc is adjacent to any fuzzy cut node then W (S ) ≥ W (S) c]. G has more than one maximum weighted arc and one of these is adjacent to a fuzzy cut node and other is adjacent to a fuzzy end node then W (S ) > W (S).
In case 4 we can consider the cases a, b, c as in case 3, we get similar results.
Therefore in all the cases we get a contradiction. Hence the minimum strong set dominating set of G is the set of all fuzzy cut nodes of G.
Hence γ ss (G) = W (S) Theorem 4.7. Let G : (V, σ, µ) be a connected fuzzy graph and S be the set of all internal nodes of any maximum spanning tree of G. Then, γ ss (G) ≤ W (S) and equality holds if G is a fuzzy tree.
Proof: Every connected fuzzy graph has at least one maximum spanning tree T and γ ss (T ) = W (S) [Theorem 4.6]. By Theorem 1.6, every strong set dominating set of T is also a strong set dominating set of G and hence γ ss (G) ≤ γ ss (T ). Hence γ ss (G) ≤ W (S) and equality holds if G is a fuzzy tree by theorem 6.7. Hence the result.

PRACTICAL APPLICATION
Consider a group of students who have close relationship with each other. Depending upon the knowledge (k) they possess, fuzzy values are assigned in the range of 0 < k < 1 for poor to intelligent student respectively. Assume the students as nodes and their relationship as arcs. A student who possesses intrinsic knowledge has relationship with other students with less knowledge than him or her, may try to disseminate his or her entire knowledge to all of them. However, his or her friends could imbibe as much as they can and act accordingly. Such a scenario can be framed as a fuzzy graph.Set domination on fuzzy graph concept can be applied for assigning group mentors in the class. Among the group of students, some could not directly interact with the teachers and may hesitate for clearing their doubts. In such situations, class can be formed into groups and a leader is assigned to each group of students who have higher knowledge than others. This will help to improve overall performance of the students. To select a leader in each group, set domination concept can be applied.

ACKNOWLEDGMENTS
Author would like to thank referees for their helpful comments.