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International Journal of Computer & Organization Trends – Volume 8 Number 1 – May 2014

The Strong Split Domination Number of Fuzzy Graphs C.Y.Ponnappan 1, P.Surulinathan2, S. Basheer Ahamed3 1

2

Department of Mathematics , Government Arts College Paramakudi , Tamilnadu, India Department of Mathematics , Lathamathavan Engineering college, Kidaripatti, Alagarkovil, Madurai625301,Tamilnadu,India.

3 Department of Mathematics, P.S.N.A. College of Engineering and Technology, Dindigul, Tamilnadu, India.

Abstract– A dominating set D of a fuzzy graph G=( σ,µ) is a strong split dominating set if the induced fuzzy subgraph H=(<VD>,σ,µ) is totally disconnected. The strong split domination number γss(G) of G is the minimum fuzzy cardinality of a strong split dominating set. In this paper we study a strong split dominating sets of fuzzy graphs and investigate the relationship of γss(G) or γss with other known parameter of G. Keywords– Fuzzy graphs, fuzzy domination, split fuzzy domination number, strong split fuzzy domination number.

I. INTRODUCTION The study of domination set in graphs was begun by Ore and Berge. Kulli V.R. et.al introduced the concept of split domination and non-split domination in graphs. Rosenfield[9] introduced the notion of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as path, cycles and connectedness. A.Somasundram and S.Somasundram [10] discussed domination in Fuzzy graphs. They defined domination using effective edges in fuzzy graph. Q.M. Mahyoub and N.D. Sonar discussed the split domination number of fuzzy graphs [8].In this paper we discuss the strong split domination number of fuzzy graph and Establish the relationship with other parameter which is also investigated.

II.PRELIMINARIES Definition:2.1[10] Let V be a finite non empty set. Let E be the collection of all two element subsets of V. A fuzzy graph G=(σ,µ) is a set with two functions σ :V→[0,1] and µ: E→[0,1] such that µ({u ,v})≤σ(u)σ(v) for all u,v  V. Definition:2.2[10] Let G=( σ,µ) be a fuzzy graph on V and V1 V. Define σ1 on V1 by σ1(u)=σ(u) for all u V1 and µ1 on the collection E1 of two element subsets of V1 by µ1({u ,v}) = µ({u ,v}) for all u,v  V1, then (σ1,µ1) is called the fuzzy subgraph of G induced by V 1 and is denoted by <V1>.

Definition:2.3[10] The order p and size q of a fuzzy graph G=( σ,µ) are defined to be p=∑uV σ(u) and q=∑(u ,v)E µ({u ,v}).

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Definition:2.4 [10] Let G=( σ,µ) be a fuzzy graph on V and DV then the fuzzy cardinality of D is defined to be ∑uD σ(u).

Definition:2.5[10] An edge e={u ,v} of a fuzzy graph is called an effective edge if µ({u ,v}) = σ(u)  σ(v).

N(u) = { vV/ µ({u ,v}) = σ(u)  σ(v)} is called the neighborhood of u and N[u]=N(u)  {u} is the closed neighborhood of u.

The effective degree of a vertex u is defined to be the sum of the weights of the effective edges incident at u and is denoted by dE(u). ∑  ( ) ( ) is called the neighborhood degree of u and is denoted by dN(u). The minimum effective degree E(G)=min{dE(u)|uV(G)} and the maximum effective degree E (G) = max{dE(u)|uV(G)}.

Definition : 2.6[10] The complement of a fuzzy graph G denoted by ̅ is defined to be ̅ = (, ) where ({ , }) = ( ) ( ) − ({ , }).

Definition : 2.7[8] A set of fuzzy vertex which cover all the fuzzy edges is called a fuzzy vertex cover of G and the minimum cardinality of a fuzzy vertex cover is called a vertex covering number of G and denoted by (G).

Definition : 2.8[8] A disconnection of a fuzzy graph G is a vertex set D whose removal results in a disconnected or a single vertex graph. The weight of D is defined to be ∑  {({ , })/ ({ , }) ≠ 0,  ,  }; the vertex connectivity of fuzzy graph G denoted by (G) is defined to be the minimum weight of disconnection in G. Definition : 2.9[8] A fuzzy edge {u,v} is called strong edge in G if ({u,v})≥({u,v}) and if ({u,v})>0 then we call v has a strong neighbor u. We call v as a fuzzy end-vertext if it has at most one strong neighbor in G. Evidently, if v is an end-vertex of G*, then v is a fuzzy end vertex of G. A fuzzy vertex x is a cut vertex if and only if there exist u and v distinct from x

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International Journal of Computer & Organization Trends – Volume 8 Number 1 – May 2014 such that x is on every strongest path from u to v. A fuzzy graph G is called block if it has no cut vertex.

Definition : 2.10[10] Let :V[0,1] be a fuzzy subset of V. Then the complete fuzzy graph on  is defined to be (,) where ({u,v})=(u)(v) for all {u,v}E and is denoted by K.

Definition : 2.11[2] Let G=(V,E) be a graph. A subset D of V is called a dominating set in G if every vertex in V-D is adjacent to some vertex in D. Definition : 2.12[10] Let G=(,) be a fuzzy graph on V. Let u,vV. We say that u dominates v in G if ({u,v})=(u)(v). A subset D of V is called a dominating set in G if for every vD, there exists uD such that u dominates v. The minimum fuzzy cardinality of a dominating set in G is called the domination number of G and is denoted by (G) or .

Definition : 2.13[4] A Dominating set D of a graph G=(V,E) is a split dominating set if the induced subgraph <V-D> is disconnected.

Definition : 2.14[4] A Dominating set D of a graph G=(V,E) is a strong split dominating set if the induced subgraph <V-D> is totally disconnected with at least two vertices.

Definition : 2.15[8] A dominating set D of a fuzzy graph G=(,µ) is a split dominating set if the induced fuzzy subgraph H=(<VD>,,µ) is disconnected.

The split domination number ( ) is the minimum fuzzy cardinality of a split domination set.

Definition : 2.16 A dominating set D of a fuzzy graph G=(σ,µ) is a strong split dominating set if the induced fuzzy subgraph H=(<VD>,σ',µ') is totally disconnected.

The strong split domination number ( ) or is the minimum fuzzy cardinality of a strong split dominating set.

Definition : 2.17[10] A fuzzy graph G=(,µ) is said to be bipartite if the vertex V can be partitioned into two nonempty sets V1 and V2 such that µ(v1,v2)=0 if v1,v2V1 or v1,v2V2. Further if (u,v)=(u)  (v) for all uV1 and vV2 then G is called a complete bipartite graph and is denoted by  , where 1 and 2 are, respectively, the restrictions of  to V1 and V2.

Definition : 2.18 [8] A dominating set D of a fuzzy graph G=(σ,µ) is connected dominating set if the induced fuzzy sub graph H=(<D>,,µ) is connected. The minimum fuzzy cardinality of a connected dominating set of G is called the connected dominating number of G and is denoted by ( ) (or) .

Definition : 2.19[10]

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A dominating set D of a fuzzy graph G is said to be a minimal dominating if no proper subset D of D is dominating set of G such that |D|<|D|.

Proposition : For any complete fuzzy bipartite graph  , where |V1|=m and |V2|=n where m=(v), vV1 and n=(v), vV 2 then ss(  , )=min{m,n}.

Theorem : 1 A dominating set D of a fuzzy graph G is a strong split dominating set if and only if there exists two fuzzy vertices v, uV-D such that every v-u path contains a fuzzy vertex of D. Proof :

u 1(0.5)

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D = {u2 , u4} V-D = {u1,u3} <V-D> is a fuzzy sub graph of G induced by <V-D> is totally disconnected, hence D is a strong split dominating set of G with strong split domination number  ( ) = 0.4+0.2 = 0.6. we see that there exists u1,u3V-D such that u1-u3 path contains u2.

Theorem : 2 For any fuzzy graph G=(,µ) (i) ( )≤ ( ) (ii) ( ) ≤ ( ) Proof : ( ),

(i) and (ii) follows from the definitions of ( ) and ( )

Theorem : 3 For any fuzzy graph G=(σ,µ)  ( ) ≤ ( ) -- (1) where (G) is a fuzzy vertex covering number of G. Proof : Let D be a maximal independent set of a fuzzy vertex in G, then D has at least two fuzzy vertices and every fuzzy vertex in D is adjacent to some vertex in V-D. This implies that V-D is a strong split dominating set of G. Thus (1) holds.

Example :

u1(0.5) 0.4

u2(0.4) 0.4

0.2 u6(0.2)

u 3(0.6) 0.5

0.2 u5(0.3)

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International Journal of Computer & Organization Trends – Volume 8 Number 1 – May 2014 D1 = {u 2, u4} ( ) = 1.0 D2 = {u 1,u 3, u5} ( ) = 1.5 D3 = {u 2, u4, u 5} = 1.5 Theorem : 4 = 1.5 A strong split dominating set D of G is minimal if D4 = {u 1,u 2, u5} D = {u2 , u4 , u6},  = 1.1 = min {1.1 , 1.4} = 1.1

and only if for each vertex D one of the following conditions holds. (i)

There exists a vertex uV-D such that N(u)D={}.  is an isolate in <D>. <V-D> is connected.

(ii) (iii)

= 1.0 = 1.0 =

Theorem : 7 ( ) ≤ ( ), then for any strong split If dominating set D of G, V-D is a strong split dominating set of G. Proof :

Proof : Suppose that D is minimal and there exists a vertex D such that  does not satisfy any of the above conditions.

Then by conditions (i) and (ii) D=D-{} is a dominating set of G, also by (iii) <V-D> is totally disconnected. This implies that D is strong split dominating set of G, which is contradiction.

Since D is minimal, by theorem (4), V-D is dominating set of G and furthermore it is a strong split dominating set since <D> is totally disconnected.

u 1(0.2) 0.2

u 2(0.3)

u6(0.4)

Theorem : 5 For any fuzzy graph G=(,µ), ( ) ≤ . ∆( )/(∆( ) + 1) - (1)

0.2

0.2 0.2

0.2 u 3(0.2) 0.2

Proof :

u5(0.5) u4(0.4) Let D be a strong split dominating set. Since D is minimal, by theorem (4) it follows that for each D there Dss={u1,u3} exist uV-D such that 0<µ(u,)=(u)() ( is adjacent to u) D ={u ,u ,u } V-D = {u2,u 4,u 5,u6} c 2 3 6 this implies that V-D is a dominating set of G. = 0.9 = 0.4 Thus ( ) ≤ | − | ≤ − ( ) and by, for any fuzzy graph G=(,) ( ) ≥ /(∆( ) + 1), hence (1) is hold.

Theorem : 8 In the next result we obtain a sufficient condition on Let G=(,µ) be a fuzzy graph such that both G and ̅ ( ) = ( ).

G such that are connected, then ( ) + ( ̅ ) ≤ 2 Theorem : 6 For any fuzzy graph G=(,µ), with fuzzy end-vertex. ( ) = ( ) futhermore, there exists a strong split dominating set of G containing all vertices adjacent to a fuzzy end-vertices.

Proof :

Proof :

Theorem : 9 If G=(,µ) has one fuzzy cut vertex v and atleast two fuzzy blocks H1 and H2 with v adjacent to all other vertices of H1 and H2, then v is in every strong split dominating set of G.

Let v be a fuzzy end vertex of G, then there exists fuzzy cut-vertex x such that µ(u,v)>0, let D be a dominating set of G, suppose that xD then D is a strong split dominating set of G. Repeating this process for all such cut-vertices adjacent to a fuzzy end-vertices, we obtain a strong split dominating set of G containing all cut-vertices adjacent to the end-vertices.

Example :

u1 0.5

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u5 0.5

By Theorem (3) ( ) ≤ ( ). Since both G and ̅ are connected, then ∆( ), ∆( ̅ ) ≤ this implies 0(G), ( ) ≤ . Similarly ( ̅ ) ≤ . Thus, 0 ( ̅ ) ≥0. Hence ( ) + ( ̅) ≤ + = 2 .

Proof : Let D be a strong split dominating set of G, suppose vV-D, then each of H1 and H2 contributes atleast one vertex to D, say u and w respectively.

This implies thatD=D-{u,w}  {v} is a strong split dominating set of G. Which is contradiction.

Hence v is in every strong split dominating set of G.

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International Journal of Computer & Organization Trends – Volume 8 Number 1 – May 2014 s(0.3) 0.3

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w(0.3) 10.

Q.M. Mahioub and N.D. Soner (2007), “The split domination number of fuzzy graph “Accepted for publication in Far East Journal of Applied Mathematics”.

Rosenfeld, A., 1975. Fuzzy graphs. In : Zadeh, L.A., Fu, K.S., Shimura, M. (Eds.), Fuzzy Sets and Their Applications. Aca-demic Press, New York.

Somasundaram, A., and Somasundaram, S., Domination in fuzzy graphs, Pattern Recognit. Lett. 19(9) 1998), 787-791.

y(0.1)

D={u,v,w} Theorem : 10 Let v be a fuzzy cut vertex of G, if there is a block H in G such that v is the only cut vertex of H and v is adjacent to all vertices of H, then there is a strong split dominating set of G containing v.

Proof : u(0.5) (0.3) v(0.3) 0.2

0.3 0.3

w(0.2) 0.2

x(0.4) 0.3

y(0.3)

D={u,v,y} If there exist atleast two blocks in G satisfying the given condition, then by theorem(9), v is in every strong split dominating set of G and hence the result.

Suppose there exists only one block H in G satisfying the given condition. Let D be a strong split dominating set of G, suppose vV-D, then for some vertex uH, {u}D. This proves that D=D-{u}{v} is a strong split dominating set of G which is contradiction.

ACKNOWLEDGEMENT Thanks are due to the referees for their valuable comments and suggestions.

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