Connectivity in Fuzzy Soft graph and its Complement

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1 IOSR Journal of Mathemats (IOSR-JM) e-issn: , p-issn: X. Volume 1 Issue 5 Ver. IV (Sep. - Ot.2016), PP Connetvty n Fuzzy Soft graph and ts Complement Shashkala S 1, Anl P N 2 1,2 (Department of Mathemats, Global Aademy of Tehnology, Bangalore, Inda) Abstrat: The onept of onnetvty plays a vtal role n the theory and applatons of graphs. In ths paper, the onept of fuzzy sets and soft sets are appled to study unertanty and vagueness n graphs. The onept of m-strong fuzzy soft ar and the omplement of fuzzy soft graph are ntrodued. Connetvty n fuzzy soft graphs n omparson wth ther omplements s dsussed through varous examples. Keywords : fuzzy graph, fuzzy soft graph, omplement of fuzzy soft graph, onnetvty n fuzzy soft graph, m- strong fuzzy soft ar. I. Introduton The onept of soft set theory was ntated by Molodtsov [1] for dealng wth unertantes. A Rosenfeld [2] developed the theory of fuzzy graphs n 1975 by onsderng fuzzy relatons on fuzzy sets, whh was developed by Zadeh [3] n the year Maj et al. [4] dsussed some applatons of fuzzy soft sets to deson makng problems. The onept of omplement of fuzzy graph was presented by M.S. Suntha and A. Vjayakumar [5]. Some operatons on fuzzy graphs were studed by Mordeson and C.S. Peng [6]. Later, Al et al. [7] dsussed about fuzzy sets and fuzzy soft sets ndued by soft sets. Varous onnetedness onepts n fuzzy graphs were ntrodued by Yeh and Bang [8]. Connetvty of fuzzy graph and ts omplement was analysed by K.R. Sandeep Narayan and M.S. Suntha [9]. Researh on fuzzy systems has been very atve and reeved muh attenton from researhers worldwde beause of ts applatons n varous dsplnes suh as pattern reognton [10], hemstry [11], neural networks [12] and deson makng [13]. M Akram and S Nawaz [14] ntrodued fuzzy soft graphs n the year Sumt Mohnta and T K Samanta [15] also ntrodued fuzzy soft graphs ndependently. The noton of fuzzy soft graph and few propertes related to t are presented n ther paper. Reently, Akram and Zafar [16] ntrodued the notons of fuzzy soft trees, fuzzy soft yles, fuzzy soft brdges and nvestgated some of ther fundamental propertes. In ths paper, m-strong fuzzy soft ar, omplement of fuzzy soft graph, onnetvty of fuzzy soft graph are ntrodued and some of ts propertes are studed. II. Prelmnares Some defntons whh an be found n [14,15,16,17] are revewed here. Defnton 1: A fuzzy graph s a par G :,, where s a fuzzy subset of a set V and s a fuzzy relaton on,.e. ( x, ( x) ( x, y. We assume that V s fnte and non-empty, s reflexve and symmetr. In all the examples s hosen sutably. Also we denote underlyng rsp graph by G :,, where u / ( u) 0 and ( u, : ( u, 0. Defnton 2: A fuzzy soft graph G ( G, F, K, s a 4-tuple suh that a) G ( V, s a smple graph b) A s a non-empty set of parameters ) ( F, s a fuzzy soft set over V d) ( K, s a fuzzy soft set over E e) ( F, e),( K, e) s a fuzzy graph of G e A.e. x mn{ x), } for all e A and, y V. by H ( e ) for onvenene. x The fuzzy graph (, e),( K, e) F s denoted G for a smple graph, G for a fuzzy soft graph and H for a fuzzy graph. In what follows, we wll use Defnton 3: G1 ( G, F1, K1, s a spannng fuzzy soft subgraph of G ( G, F, K, f both have the same fuzzy soft vertex set.e. F1 ( e) e) e A. They dffer only n the ar weghts. H ( e ) F ( e ), K ( e ) Defnton 4: A fuzzy soft graph G s a fuzzy soft tree f eah fuzzy graph DOI: / Page

2 Connetvty n Fuzzy Soft graph and ts Complement e A has a fuzzy spannng subgraph Q ( e ) ( ), F e T ( e ) whh s a tree, where for all ars xy not n Q ( e ), e )( x CONN ( x. ( ) Q( e ) CONN Q ( e ) xy denotes the strength of onnetedness between two nodes x and y whh s defned as the maxmum of strengths of all paths between x and y. We assume that eah node has membershp value 1 unless otherwse spefed throughout the examples of ths paper. In all odd numbered fgures, A and B denotes fuzzy graphs orrespondng to parameter e 1 and respetvely n G and n all even numbered fgures, A and B denotes fuzzy graphs orrespondng to parameter e 1 and respetvely n G. III. Complement Of Fuzzy Soft Graph And M-Strong Ar In Fuzzy Soft Graph Defnton 5: The omplement of a fuzzy soft graph G s a fuzzy soft graph denoted by set over V s same n both G and Defnton 6: An ar (, strong, then K ( u 0 G where the fuzzy soft G and K ( x mn{ x), } x e A, x, y. u mn{ u), v for some e A. If ( u, s u of G s alled m-strong f )} Example 1) Consder a smple graph G ( V, suh that V { a1, a a3} and E { a1a2, a2a3, a3a1}. Let A { e 1, } be a parameter set and ( F, be a fuzzy soft set over V wth ts fuzzy approxmate funton F : A P( V ) defned by ) {( a1, 0.3),( a 0.7),( a3, 0.5)} ) {( a1, 0.4),( a 0.3),( a3, 0.8)} Let ( K, be a fuzzy soft set over E wth ts fuzzy approxmate funton K : A P( defned by ) {( a1a 0.3),( a2a3, 0.4),( a3a1, 0.2)} e ) {( a a, 0.2),( a a, 0.1),( a, 0.3)} Fg.1 Fuzzy Soft Graph G Fg.2 Fuzzy Soft graph omplement DOI: / Page G

3 Connetvty n Fuzzy Soft graph and ts Complement Thus, H( ) ( ), )), H( ) ( ), )) are fuzzy graphs. Hene, G { H ( ), H ( e 2 )} s a fuzzy soft graph as shown n Fg. 1 and ts omplement G { H ( ), H ( e 2 )} as shown n Fg. 2. Here, a1a2 s an m-strong ar n G. 3.1 Connetvty n G Defnton 7: Let G be a fuzzy soft graph of. G G s sad to be a onneted fuzzy soft graph f eah H ( e ) s a onneted fuzzy graph e A..e. CONN( x, 0 x, y for eah fuzzy graph n G. There may be ases n whh the omplement of onneted fuzzy soft graphs beomes dsonneted. In ths paper, we propose a rteron by whh a fuzzy soft graph and ts omplement wll be onneted smultaneously. We llustrate these wth few examples. Proposton: If G s onneted fuzzy soft graph wth no m-strong ars then G s onneted. Proof: Consder a fuzzy soft graph G whh s onneted and ontan no m-strong ars. Let A be the set of parameters and ( F, be a fuzzy soft set over V. By the defnton, over V as defned n G. Sne G s onneted, G also ontans the same fuzzy soft set CONN( x, 0 x, y for every fuzzy graph n G. Let the path of x and y be denoted by P..e. P ( r0, r1 )( r1, r2 )...( r n 1, rn ) where r0 x, rn y and K ( e )( r 1, r ) 0. Sne G ontan no m-strong ars, K ( e )( r 1, r ) 0 n G. Hene P wll also be a path n G. Therefore, G s onneted. But, the onverse of the above may or may not be true as shown n the followng examples. Example 2) Consder a smple graph G ( V, suh that V { a1, a a3} and E { a1a2, a2a3, a3a1}. Let A { e 1, } be a parameter set and ( F, be a fuzzy soft set over V wth ts fuzzy approxmate funton F : A P( V ) defned by ) {( a1,1),( a1),( a3,1)} ) {( a1, 1),( a 1),( a3, 1)} Let ( K, be a fuzzy soft set over E wth ts fuzzy approxmate funton K : A P( defned by ) {( a1a 1),( a2a3, 0.3),( a3a1, 0.2)} e ) {( a a, 0.3),( a a, 0.1),( a, 1)} Fg.3 Fuzzy Soft Graph G Fg.4 Fuzzy Soft graph omplement DOI: / Page G

4 Connetvty n Fuzzy Soft graph and ts Complement Thus, H( ) ( ), )), H( ) ( ), )) are fuzzy graphs. Here, a 2 s an m-strong ar orrespondng to parameter n fuzzy soft graph G and stll G s onneted as shown n Fg. 3 and Fg. 4. Example 3) Consder the prevous example wth the same set of parameter, fuzzy soft set defned over V n a fuzzy soft graph G and the fuzzy soft set over E defned by ) {( a1a 1),( a2a3, 0.3),( a3a1, 1)} e ) {( a a, 0.3),( a a, 0.2),( a, 0.1)} 1a Fg.5 Fuzzy Soft Graph G Fg.6 Fuzzy Soft graph omplement Here, a1a2 and a1a3 are two m-strong ars orrespondng to parameter n G but dsonneted as shown n Fg. 5 and Fg. 6. Theorem: Let G be a fuzzy soft graph. G and DOI: / Page G G s G are onneted f and only f G ontans atleast one onneted spannng fuzzy soft subgraph wth no m-strong ars.proof: Let us assume that G ontans a spannng fuzzy soft subgraph G1 that s onneted, wth no m-strong ars. Usng proposton, sne G1 ontans no m-strong ars and s onneted, G 1 wll be a onneted spannng fuzzy soft subgraph of G and thus G s also onneted. Conversely, let us assume that G and G are onneted. We have to fnd a onneted spannng fuzzy soft subgraph of G that ontans no m-strong ars.let G1 be an arbtrary onneted spannng fuzzy soft subgraph of G. If G1 ontan no m-strong ars then G1 s the requred subgraph. Suppose G1 ontan one m- strong ar say ar ( u, n a fuzzy graph H ( e ) for some e A. Then ar ( u, wll not be present n ( e H ) n G. Let ths path be P. 1 Let P1 ( u1, u2)( u u3)...( u n 1, un ) where u1 u and un v. If all the ars of P1 are present n H ( e ) n G then H( e ) ( u, together wth P1 wll be the requred spannng subgraph. If not, then there exsts atleast one ar say ( u1, v1 ) n P1 whh s not n H ( e ) n G. Sne G s onneted, replae ( u1, v1 ) by another u1 v1 path n H( e ). Let ths path be denoted by P2. If P2 ontan no m-strong ars then H( e ) ( u, ( u 1, v 1 ) together wth P1 and P2 wll be the requred spannng subgraph. If P2 ontan an m-strong ar then ths ar wll not be present n

5 Connetvty n Fuzzy Soft graph and ts Complement H ( ) n G. Replae ths ar by a path onnetng the orrespondng vertes n H ( ) n e e G and proeed as above. Sne G ontan only fnte number of ars, atlast we wll get a spannng subgraph that ontan no m-strong ars. If more than one m-strong ar s present n G 1, then the above proedure an be repeated for all other m-strong ars of G1 to get the requred spannng subgraph of G. Corollary: Let G be a fuzzy soft graph. G and G are onneted f and only f G ontans atleast one fuzzy soft spannng tree havng no m-strong ars. Proof: From prevous theorem, we get a onneted fuzzy soft spannng subgraph of G whh ontan no m-strong ars. The maxmum spannng fuzzy soft tree of ths subgraph wll be a spannng fuzzy soft tree of G that ontans no m-strong ars. IV. Conluson In Reent Tmes, Fuzzy Soft Graph Is Fndng Applatons In The Feld Of Computer Sene In Deson Makng Problems. In Ths Paper, We Propose A Crteron To Determne The Connetvty Of Fuzzy Soft Graph And Its Complement Based On M-Strong Ars. We Have Shown That If Is A Conneted Fuzzy Soft Graph Wth No M-Strong Ars Then Is Conneted. It Is Demonstrated Wth Examples That If Is A Conneted Fuzzy Soft Graph Wth One M-Strong Ar Then Remans Conneted And If Is A Conneted Fuzzy Soft Graph Wth Two M-Strong Ars Then Is Dsonneted. We Have Also Establshed That And Are Conneted If And Only If Contans Atleast One Conneted Spannng Fuzzy Soft Subgraph Wth No M-Strong Ars. Referenes [1]. Dmtry Molodtsov, Soft set theory frst results, Computers & Mathemats wth Applatons, 37 (4), 1999, [2]. A. Rosenfeld, Fuzzy graphs, n: L A Zadeh, K S Fu, M Shmura (eds), Fuzzy sets and ther Applatons( New York : Aadem press 1975) [3]. Lotf A Zadeh, Fuzzy sets, Informaton and ontrol, 8(3),1965, [4]. P. K. Maj, Akhl Ranjan Roy, and Ranjt Bswas, An applaton of soft sets n a deson makng problem, Computers & Mathemats wth Applatons, 44 (8), [5]. M S Suntha and A Vjayakumar, Complement of a fuzzy graph, Indan Journal of Pure and Appled Mathemats, 33(9), [6]. John N Mordeson and Peng Chang-Shvh, Operatons on fuzzy graphs, Informaton Senes, 79 (3-4), 1994, [7]. Muhammad Irfan Al, A note on soft sets, rough soft sets and fuzzy soft sets, Appled Soft Computng, 11 (4), 2011, [8]. Raymond T Yeh and S Y Bang, Fuzzy relatons, fuzzy graphs and ther applatons to lusterng analyss, Fuzzy sets and ther Applatons, 1975, [9]. K R Sandeep Narayan and M S Suntha,, Connetvty n a fuzzy graph and ts omplement, Gen, 9(1), [10]. Sott C Newton, Surya Pemmaraju and Sunanda Mtra, Adaptve fuzzy leader lusterng of omplex data sets n pattern reognton, IEEE Transatons on Neural Networks, 3(5), [11]. J Xu, The use of fuzzy graphs n hemal struture researh, Fuzzy Log n Chemstry, 1997, [12]. James J Bukley and Yoh Hayash, Fuzzy neural networks: A survey, Fuzzy sets and systems, 66 (1), 1994, [13]. Mar Roubens, Fuzzy sets and deson analyss, Fuzzy sets and systems, 90(2), 1997, [14]. Muhammad Akram and Sara Nawaz, On fuzzy soft graphs, Italan Journal of Pure and Appled Mathemats, 34, 2015, [15]. Sumt Mohnta and T. K. Samanta, An ntroduton to fuzzy soft graph, Mathemata Morava, 19 (2), 2015, [16]. Muhammad Akram and Farha Zafar, Fuzzy soft trees, Southeast Asan Bulletn of Mathemats, 40 (2), 2016, [17]. John N Mordeson and Premhand S Nar, Fuzzy graphs and fuzzy hypergraphs, Physa, 46, DOI: / Page