Bridges and cut-vertices of Intuitionistic Fuzzy Graph Structure

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1 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A Brdges and cut-vertces of Intutonstc Fuzzy Graph Structure P.K. Sharma * Vandana Bansal ** Abstract Key words: Intutonstc fuzzy graph structure; Brdges; -Cut-vertces. In ths paper,, the concept of brdge and cut vertces n an ntutonstc fuzzy graph structures (IFGS) are defned and ther propertes are studed. We descrbe the exstence of brdge n an IFGS and obtan some equvalent condtons. Also ntutonstc fuzzy brdges and ntutonstc fuzzy cut vertces are characterzed usng partal ntutonstc fuzzy spannng subgraph structures... Author Correspondence: ** Vandana Bansal, Correspondng Author, RS, IKGPT Unversty, Jalandhar; Assocate Professor, RG College, Phagwara, Punjab, Inda; 2010 Mathematcs Subject Classfcaton: 03F55, 05C05, 05C72, 05C Introducton: The dea of fuzzy sets was orgnated by L.A. Zadeh [14] n A. Rosenfeld [9] commenced * Assocate Professor, P.G. Department of Mathematcs, D.A.V. College, Jalandhar, Punjab, Inda. ** Correspondng Author, Assocate Professor, RG College, Phagwara; RS, IKGPT Unversty, Jalandhar, Punjab, 349 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

2 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A the dea of fuzzy relaton and fuzzy graph and developed the structure of fuzzy graphs, obtanng analogs of several graph theoretcal concepts. The noton of graph G = (V, E) to graph structure G = (V, R 1, R 2,..., R k ) was generalzed by E. Sampatkumar n [11]. The overvew of fuzzy graph structure was later dscussed by T. Dnesh and T. V. Ramakrshnan [2]. M. G. Karunambga, O. K. Kalavan n [3] defned the brdge of IFG. Shek Dhavudh, R. Srnvasan n [10] dscussed the cutvertces of IFG.. 2. Prelmnares: In ths secton, we revew some defntons that are necessary n ths paper whch are manly taken from [2], [3], [11], [12] and [13]. Defnton (2.1): Let G = (V,R 1,R 2,...,R k ) be a graph and let A be an ntutonstc fuzzy subset on V and B 1, B 2,...,B k are ntutonstc fuzzy relatons on V whch are mutually dsjont symmetrc and rreflexve respectvely such that ( u, v ) µ u µ v and ( u, v) ( u) v u, v V and = 1,2,..., k. B A A B A A Then = (A, B 1,B 2,..,B k ) s an ntutonstc fuzzy graph structure of G. Defnton (2.2): Let = (A, B 1,B 2,...,B k ) be an ntutonstc fuzzy graph structure of a graph structure G=(V,R 1,R 2,...,R k ), then =(A,C 1,C 2,...,C k ) s called a partal ntutonstc fuzzy spannng subgraph structure of =(A,B 1,B 2,...,B k ) f ( u, v) ( u, v) and ( u, v) ( u, v) for r =1,2,...,k and u,v V, uv and =1,2,..., k. Cr Br Cr Br Note(2.3): Throughout ths paper, unless otherwse specfed = (A,B 1,B 2,...,B k ) wll represent an ntutonstc fuzzy graph structure wth respect to graph structure G = (V, R 1,R 2,...,R k ) and, for =1, 2,..., k wll refer to the number of ntutonstc fuzzy relatons on V. Defnton (2.4): Let be an IFGS of a graph structure G. If (u,v) supp( )= { (u,v) V V : (u,v) 1}, then (u,v) s sad to be a edge of. (u,v) 0, Defnton (2.5): In an IFGS, -path s a sequence of vertces u 0,u 1,...,u n whch are dstnct (except possbly u 0 = u n ) such that (u j 1,u j ) s a -edge for all j = 1,2,...,n. Defnton (2.6): In an IFGS,a path s a sequence of vertces v 1,v 2,..., v n (V ) whch are dstnct (except possbly v 1 = v n ) such that (v j, v j+1 ) s a B j -edge for some j = 1,2,, n and = 1,2,3,..., k. 350 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

3 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A Defnton (2.7): In an IFGS, the -strength of a B -path u 0,u 1,...,u n s denoted by S and s the mn (u B j 1,u j ) for j=1,2,,n..e. S = n j1 (u j 1,u j ) for =1,2,...,k. B Defnton (2.8): In an IFGS, the -strength of a B -path u 0,u 1,...,u n s denoted by S and s the max (u B j 1,u j ) for j=1,2,,n..e. S = n j1 (u j 1,u j ) for =1,2,...,k. Defnton (2.9): The strength of a -path u 0,u 1,...,u n n an IFGS ( n j1 (u B j 1,u j ), n j1 (u j 1,u j ) ) for =1,2,...,k. s denoted by S and s defned as S = Defnton (2.10): The strength S of a path n an IFGS k k of path = S mn S,max B SB 1 1. s the weght of the weakest edge of the path..e., strength Defnton (2.11): In any IFGS, 2 (u,v) = j B (u,v) = Max{ (u,w) B (u,v)= ( j 1 B B B (w,v)}and )(u,v), j=2,3,...,m for any m 2. Also (u,v) = j1 j B (u,v). Defnton (2.12): In any IFGS, 2 (u,v) = B (u,v)= Mn{ (u,w) (w,v)}and B (u,v) = ( j B ) (u,v), j=2,3,...,m for any m 2. j 1 B B Also (u,v) = j1 (u,v). j B Defnton (2.13): In an IFGS, a -cycle s an alternatng sequence of vertces and edges u 0,e 1,u 1,e 2,...,u n 1,e n,u n = u 0 consstng only of -edges. 351 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

4 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A Defnton (2.14): An IFGS s a -forest f the subgraph structure nduced by -edges s a forest,.e., f t has no -cycles. Result (2.15): s a -tree when t s a connected forest. Defnton (2.16): s an ntutonstc fuzzy -forest f t has a partal ntutonstc fuzzy spannng sub-graph structure = (A,C 1,C 2,...,C k ) whch s a C -forest where for all -edges not n H, (u,v) < (x,y). C (x,y) < (x,y) C and Defnton (2.17): s an ntutonstc fuzzy -tree f t has a partal ntutonstc fuzzy spannng sub-graph structure = (A,C 1,C 2,...C k ) whch s a C -tree where for all -edges not n, (x,y) < (x,y) and C (u,v) < (x,y). C Theorem (2.18): Let be a -cycle. s an ntutonstc fuzzy -cycle ff s not an ntutonstc fuzzy - tree. 3. -Brdges and -Cut-vertces of IFGS Defnton (3.1): An edge (u,v) s sad to be a -brdge n an IFGS f ether (u,v) < (u,v) and (u,v) (u,v) or (u,v) (u,v) and (u,v) > (u,v). In other words, deletng an edge (u,v) reduces the -strength of connectedness between some par of vertces or (u,v) s a brdge f there exsts vertces x and y s.t. (u,v) s an edge of every strongest path from x to y. Defnton (3.2): If an IFGS has at least one -brdge, s sad to have a brdge. Theorem (3.3): () If there exsts one, ( =1,2,..,k ) whch s constant then has no -brdge. () If there exsts one, ( =1,2,..,k ) whch s not constant then has a -brdge. Proof: () Suppose that all ( = 1, 2,., k ) are constant. Let (u, v) = c and (u, v) = d u, v V where 0 c, d 1. Snce the degree of membershp of each edge are same (.e., c) and degree of non- membershp of each edge are also same (.e., d). Therefore, deletng any edge does not reduce the strength of connectedness between any par of vertces. Hence has no -brdge. 352 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

5 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A () Assume that s not constant. Choose an edge (u x, v x )V V such that (u x, v x ) = max { (u, v) : (u, v) V V} and (u B x, v x ) = mn { (u, v) : (u, v) V V }. Snce (u B x,v x ) 0 and (u B x,v x ) < 1 therefore, there exsts atleast one - edge (u y, v y ) dstnct from (u x,v x ) such that (u B y,v y ) (u B x,v x ) and (u B y,v y ) > (u B x,v x ). If we delete the -edge (u x,v x ), then the strength of connectedness between u x and v x n the fuzzy subgraph structure thus obtaned s decreased..e., ( u B x,v x ) < (u x,v x ) and ( u x,v x ) > (u B x,v x ). (u x,v x ) s a brdge of. ( by defnton of brdge.) Theorem (3.4): In an IFGS = (A, B 1,B 2,...,B k ), after deletng a -edge (u,v), we have an IFGS = (A,B 1,B 2,,B k ) of vertces (u x,v x ) for (x,y =1,2,...,n) then the followng condtons are equvalent: () (u,v)< (u,v) and () (u,v) s a -brdge. (u,v) > (u,v). () (u,v) s a not a -edge of any cycle. Proof: To Prove () (). Gven that (u,v)< (u,v) and To prove (u,v) s a -brdge. (u,v) > (u,v). Suppose that (u,v) s not a -brdge, then (u x,v y ) = (u,v) (u,v) and whch contradcts (). (u,v) s a -brdge. To Prove () (). Gven (u,v) s a -brdge. To Prove (u,v) s a not a -edge of any cycle. Suppose (u,v) s a a -edge of any cycle, ( u x,v y ) = (u,v) (u,v) any path whch has a -edge (u,v) wth the use of rest of the cycle as a path from u to v whch s a contradcton to our assumpton. (u,v) s a not a -edge of any cycle. 353 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

6 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A To Prove () (). Let (u,v) s a not a -edge of any cycle. To Prove Suppose (u,v)< (u,v) and (u x,v y ) (u,v) and (u,v) > (u,v). ( u x,v y ) (u,v). Then there exsts a path from u to v whch does not nvolve (u,v) that has strength greater than or equal to (u,v) and less than or equal to (u,v). Also ths path together wth (u,v) form a cycle, whch s a contradcton to our assumpton. (u,v)< (u,v) and (u,v) > (u,v). Hence (), () and () are equvalent. Theorem (3.5): If (u,v) s a -brdge of an IFGS G = (A, B 1,B 2,...,B k ) and H = (A,B 1,B 2,,B k ) s a partal ntutonstc fuzzy spannng subgraph structure obtaned by deletng (u,v) for =1,2,...,k. Then and (u,v) > (u,v). (u,v) < (u,v) Proof: If possble, Suppose there exsts a -path of strength greater than (u,v) and less than (u,v) from u to v not havng the -edge (u,v)..e., suppose u, v u, v and u, v u, v. B B B B Any -path whch contans -edge (u,v) can be replaced by a -path whch does not have -edge (u,v) and ts strength s not reduced. Ths contradcts that (u,v) s a -brdge of Thus (u,v) < (u,v) and (u,v) > (u,v) for =1,2,...,k. Corollary (3.6): Converse of the above theorem s also true..e., f (u,v) < then (u,v) s a -brdge of (u,v) and (u,v) > (u,v), Theorem (3.7): Let be an ntutonstc fuzzy graph structure whch s an ntutonstc fuzzy -forest. Then the edge of the partal ntutonstc fuzzy spannng subgraph structure the -brdges of. Proof: Two cases arses. Case I: (u,v) s a -edge whch does not belong to H. By defnton of an ntutonstc fuzzy -forest, H = (A, C 1, C 2,...,C k ) whch s a C - forest, are 354 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

7 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A (u,v)< (u,v) (u,v) C and (u,v) > (u,v) C ntutonstc fuzzy spannng subgraph structure obtaned by deletng (u,v). by theorem (3.5), (u,v) s not a -brdge. Case II: (u,v) s a C -edge whch belongs to H If possble, suppose (u,v) s not a -brdge, (u,v) where (A,B B 1,B 2,,B k ) be a partal there exsts a -path P from u to v not havng (u,v) wth strength greater than or equal to (u,v) and less than or equal to (u,v). u, v = u, v u, v and u, v u, v u, v. B B B B B B P and H form B -cycle. But H does not contan C -cycle, P contans -edge not n Let (x,y) be a -edge of P. H. By defnton of an ntutonstc fuzzy -forest, t can be replaced by a C -path n than (x,y) and less than (x,y). Also x y x y, u, v and, u, v. B B B B H whch has strength greater All C -edges of P are stronger than (x,y) and equal to (u,v). Thus P does not have (u,v). If t contans (u,v), ts strength wll be less than or equal to u, v u, v and u, v u, v C B C B there exsts a C -path n H from u to v not havng (u,v). there exsts a C -cycle n H. And thus there exsts a -cycle whch s not possble. (u,v) s a -brdge. Hence edge of H are the B -brdges of.. (x,y) whch s greater than or equal to (u,v) and less than or (u,v) and greater than or equal to (u,v),.e., Defnton (3.8): =(A 1,B 1,B 2,...,B k ) s the partal ntutonstc fuzzy subgraph structure obtaned by removng a vertex w of,.e., 355 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

8 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A (w) = 0 and A 1 (u) = (u) u w, A 1 A 1 (w,v) =0 and (w,v) = 0 v V and (u,v) = (u,v) and (u,v) = (u,v) (u,v) (w,v), =1,2,...,k. Defnton (3.9): A vertex w of between some par of vertces. s a -cut vertex f deletng t reduces the - strength of connectedness Defnton (3.10): A vertex w of between some par of vertces. s a -cut vertex f deletng t reduces the - strength of connectedness Defnton (3.11): A vertex w s sad to be a cut vertex of ntutonstc fuzzy graph structure vertex w reduces the - strength of connectedness between some par of vertces. In other words, f ether f deletng a (u,v) < (u,v) and (u,v) (u,v) or (u,v) (u,v) and (u,v) > (u,v) for some u,v V. Now we dscuss some results on -brdges and -cut vertces. Theorem (3.12): Let be an IFGS wth *= (supp(a), supp(b 1 ), supp(b 2 ),..., supp(b k )) a -cycle. If a vertex of s a cut vertex of, then t s a a common vertex of two -brdges. Proof: Consder a -cut vertex w of. By the defnton of a -cut vertex, there exsts two vertces u and v dfferent from w such that w s on every strongest u v -path. Gven that s a -cycle. then there exsts only one strongest - path P from u to v contanng w. All -edges of P are -brdges. So w s common to two -brdges. Converse of the above result s also true as s apparent from the next theorem: Theorem (3.13): Let be an IFGS. If w s common to at least two -brdges of, then w s a -cut vertex. Proof: Let (u 1,w) and (w, v 2 ) be two -brdges wth w as the common vertex. Snce (u 1,w) s a -brdge, t s on every strongest u-v -path for some u and v. Case I: w u, w v In ths case, w s on every strongest u -v -path for some u and v. Then w s a -cut vertex. Case II: Ether w = u or w = v In ths case ether (u 1,w) s on every strongest u -w -path or (w, v 2 ) s on every strongest w- v -path. If possble, let w be not a -cut vertex. By defnton of -cut vertex, there exsts a strongest -path not contanng w between any par of vertces. Consder such a path P jonng u 1 and v 2. Then P,(u 1,w), (w, v 2 ) form a -cycle. 356 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

9 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A Subcase (): Let u 1,w, v 2 be not a strongest -path. Then (u 1,w) or (w,v 2 ) or both become the weakest -edges of the above cycle consstng of P,(u 1,w) and (w, v 2 ) snce every -edge of P wll be stronger than (u 1,w) and (w, v 2 ). Ths s not possble snce (u 1,w) and (w, v 2 ) are -brdges. Subcase (): Let u 1, w,v 2 also be a strongest -path jonng u 1,v 2 (u B 1, v 2 ) = (u 1,w) (w,v 2 ) and (u 1, v 2 ) = (u 1,w) (w,v B 2 ).e., ether (u 1,w) or (w, v 2 ) or both are the weakest -edges of the above -cycle because P s as strong as u 1,w, v 2. Ths s not possble because u 1,w, v 2 s a strongest -path. Therefore, w s a -cut vertex. Now we prove that the nternal vertces of a -tree of an IF -tree are the -cut vertces. Theorem (3.14):Let be an ntutonstc fuzzy -tree for whch = (A,C 1,C 2,...,C k ) s a partal IF spannng subgraph structure whch s a C -tree and (x,y) < (x,y) and C (u,v) > (x,y) (x,y) not n C. Then the nternal vertces of are precsely the cut vertces of. Proof: Consder a vertex w of. Case I: w s not an end vertex of I, Therefore, w s common to two C -edges of at least and by Theorem (3.7), they are -brdges of. Then by Theorem (3.13), w s a -cut vertex. Case II: w s an end vertex of If w s a -cut vertex, t les on every strongest -path and hence C -path jonng u and v for some u and v n V. One of such C -paths les n. But w s an end vertex of. So ths s not possble. So w s not a -cut vertex.e., the nternal vertces of are precsely the -cut vertces of. The above theorem leads us to the followng corollary. Corollary (3.15): A -cut vertex of an ntutonstc fuzzy graph structure whch s an ntutonstc fuzzy - tree, s common to at least two -brdges. Acknowledgement: The second author would lke to thank IKG PT Unversty, Jalandhar for provdng an opportunty to do research work under her supervsors. 357 Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com

10 Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: Emal: Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal - Included n the Internatonal Seral Drectores Indexed & Lsted at: Ulrch's Perodcals Drectory, U.S.A., Open J-Gage as well as n Cabell s Drectores of Publshng Opportuntes, U.S.A References [1]. Atanassov, K.T, Intutonstc Fuzzy sets: Theory and applcatons, Studes n fuzzness and soft computng, Hedelberg, New York, Physca-Verlag, [2]. Dnesh T.and Ramakrshnan T. V., On Generalsed fuzzy Graph Structures, Appled Mathematcal Scences, Vol. 5, no. 4, 2011, [3]. M. G. Karunambga, O. K. Kalavan, Self centered IFG, World Appled Scences Journal 14 (12), 2011, [4]. M.G.Karunambga, R. Parvath, R. Buvaneswar, Constant ntutonstc fuzzy graphs, NIFS 17 (2011), 1, [5]. Mordeson John N., Nar Premchand S., Fuzzy Mathematcs: An Introducton for Engneers and Scentsts, Sprnger- Verlag Company, [6]. Mordeson, J.N. & Nar, P.S., Fuzzy Graphs and Fuzzy Hypergraphs, Physca-verlag, [7]. Parvath R, Karunambga M. G., Intutonstc fuzzy Graphs, Journal of Computatonal Intellgence, Theory and Applcatons, 20, 2006, [8]. Parvath R. and M.G. Karunambga and Atanassov, K.T, Operatons on Intutonstc Fuzzy Graphs, Fuzzy Systems, 2009, Proceedngs of FUZZ-IEEE 2009, South Korea, 2009, [9]. Rosenfeld A., Fuzzy Graphs, Fuzzy Sets and ther Applcatons to Cogntve and Decson Process n: L.A. Zadeh, K.S. Fu. M. Shmura (Eds), Academc Press, New York,1975, [10]. S. Shek Dhavudh, R. Srnvasan, Propertes of IFGs of second type, Internatonal Journal of Computatonal and Appled Mathematcs, Vol. 12, no. 3, 2017,pp [11]. Sampatkumar E., Generalzed Graph Structures, Bulletn of Kerala Mathematcs Assocaton, Vol. 3, No.2, (Dec 2006,), [12]. Sharma P.K. and Bansal Vandana, On Intutonstc Fuzzy Graph Structures, IOSR Journal of Mathematcs, (IOSR-JM), Vol. 12, Issue 5 Ver. I,Sep. - Oct. 2016, pp [13]. Sharma P.K. and Bansal Vandana, Some elementary operatons on Intutonstc Fuzzy Graph Structures, Internatonal Journal for Research n Appled Scence & Engneerng Technology (IJRASET), Volume 5, Issue VIII, August 2017, pp [14]. Zadeh L.A., Fuzzy Sets, Informaton and Control, 8, 1965, Internatonal Journal of Engneerng, Scence and Mathematcs Emal: jesmj@gmal.com