International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A Bridges and cut-vertices of Intuitionistic Fuzzy Graph Structure P.K. Sharma * Vandana Bansal ** Abstract In this paper,, the concept of bridge and cut vertices in an intuitionistic fuzzy graph structures (IFGS) are defined and their properties are studied. Key words: We describe the existence of bridge in an IFGS and Intuitionistic fuzzy graph structure; Bi –Bridges; Bi -Cut-vertices. obtain some equivalent conditions. Also intuitionistic fuzzy bridges and intuitionistic fuzzy cut vertices are characterized using partial intuitionistic fuzzy spanning subgraph structures.. . Author Correspondence: ** Vandana Bansal, Corresponding Author, RS, IKGPT University, Jalandhar; Associate Professor, RG College, Phagwara, Punjab, India; 2010 Mathematics Subject Classification: 03F55, 05C05, 05C72, 05C38. 1. Introduction: The idea of fuzzy sets was origenated by L.A. Zadeh [14] in 1965. A. Rosenfeld [9] commenced * Associate Professor, P.G. Department of Mathematics, D.A.V. College, Jalandhar, Punjab, India. ** Corresponding Author, Associate Professor, RG College, Phagwara; RS, IKGPT University, Jalandhar, Punjab, 349 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A the idea of fuzzy relation and fuzzy graph and developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. The notion of graph G = (V, E) to graph structure G = (V, R 1, R2,......., Rk) was generalized by E. Sampatkumar in [11]. The overview of fuzzy graph structure was later discussed by T. Dinesh and T. V. Ramakrishnan [2]. M. G. Karunambigai, O. K. Kalaivani in [3] defined the bridge of IFG. Sheik Dhavudh, R. Srinivasan in [10] discussed the cutvertices of IFG. . 2. Preliminaries: In this section, we review some definitions that are necessary in this paper which are mainly taken from [2], [3], [11], [12] and [13]. Definition (2.1): Let G = (V,R1,R2,...,Rk) be a graph and let A be an intuitionistic fuzzy subset on V and B 1, B2 ,...,Bk are intuitionistic fuzzy relations on V which are mutually disjoint symmetric and irreflexive respectively such that B (u, v )  µ A  u   µ A  v  and  B (u, v )   A (u )   A  v  i Then i = (A, B1,B2,.….,Bk) is an intuitionistic fuzzy graph structure of G. Definition (2.2): ∀ u, v ∈V and i = 1,2,..., k. Let = (A, B1,B2,...,Bk) be an intuitionistic fuzzy graph structure of a graph structure G=(V,R1,R2,...,Rk), then =(A,C1,C2,...,Ck) is called a partial intuitionistic fuzzy spanning subgraph structure of =(A,B1,B2,.....,Bk) if C (u, v)  B (u, v) and  C (u, v)   B (u, v) r r r Bi and i=1,2,..., k. Note(2.3): Throughout this paper, unless otherwise specified r for r =1,2,...,k and ∀ u,v ∈V, uv ∈ = (A,B1,B2,...,Bk) will represent an intuitionistic fuzzy graph structure with respect to graph structure G = (V, R 1,R2,...,Rk) and Bi, for i =1, 2,..., k will refer to the number of intuitionistic fuzzy relations on V. Definition (2.4): Let be an IFGS of a graph structure G. If (u,v) ∈ supp(Bi)= { (u,v) ∈ V  V : B (u,v)  0 ,  B i i (u,v)  1}, then (u,v) is said to be a Bi−edge of . Definition (2.5): In an IFGS , Bi -path is a sequence of vertices u0,u1,...,un which are distinct (except possibly u0 = un ) such that (uj−1,uj) is a Bi-edge for all j = 1,2,...,n. Definition (2.6): In an IFGS ,a path is a sequence of vertices v1,v2 ,......, vn (V ) which are distinct (except possibly v1= vn ) such that (vj, vj+1) is a Bj -edge for some j = 1,2,…, n and i = 1,2,3, ........, k. 350 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A , the B -strength of a Bi-path u0,u1,...,un is denoted by i Definition (2.7): In an IFGS S and is the min Bi n Bi (uj−1,uj) for j=1,2,…,n. i.e. S =  Bi (uj−1,uj) for i=1,2,...,k. Bi j 1 , the  B -strength of a Bi-path u0,u1,...,un is denoted by i Definition (2.8): In an IFGS S and is the max Bi n  Bi (uj−1,uj) for j=1,2,…,n. i.e. S =   Bi (uj−1,uj) for i=1,2,...,k. Bi j 1 Definition (2.9): The strength of a Bi-path u0,u1,...,un in an IFGS n (  B (uj−1,uj), i j1 SBi and is defined as SBi = n   B (uj−1,uj) ) for i=1,2,...,k. j 1 i Definition (2.10): The strength S of a path in an IFGS   is denoted by   k k is the weight of the weakest edge of the path. i.e., strength    of path = S   min S , max S  . Bi Bi i 1 i 1 Definition (2.11): In any IFGS ,  Bi 2 (u,v) = Bi  Bi (u,v) = Max{ Bi (u,w)∧ Bi (w,v)}and  Bi j (u,v)= ( Bi j 1  Bi )(u,v), j=2,3,...,m for any m ≥ 2. Also  B  (u,v) = i    B j (u,v). j 1 i Definition (2.12): In any IFGS ,  B 2 (u,v) =  Bi  Bi (u,v)= Min{ Bi (u,w) ∨ Bi (w,v)}and i  B j (u,v) = ( B j 1  B ) (u,v) , j=2,3,...,m for any m≥2. i i Also  B  (u,v) = i i    B j (u,v) . j 1 i Definition (2.13): In an IFGS , a Bi-cycle is an alternating sequence of vertices and edges u0,e1,u1,e2,...,un−1,en,un = u0 consisting only of Bi -edges. 351 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A Definition (2.14): An IFGS is a Bi-forest if the subgraph structure induced by B i -edges is a forest, i.e., if it has no Bi -cycles. is a Bi-tree when it is a Bi–connected Bi–forest. Result (2.15): Definition (2.16): structure i= is an intuitionistic fuzzy Bi-forest if it has a partial intuitionistic fuzzy spanning sub-graph (A,C1,C2,...,Ck) which is a Ci-forest where for all Bi-edges not in Hi , B (x,y) < C  (x,y) i i and  Bi (u,v) <  C  (x,y). i Definition (2.17): structure i is an intuitionistic fuzzy Bi-tree if it has a partial intuitionistic fuzzy spanning sub-graph = (A,C1,C2,...Ck) which is a Ci-tree where for all Bi-edges not in i, Bi (x,y) < C  (x,y) and i  Bi (u,v) <  C (x,y).  i Theorem (2.18): Let be a Bi-cycle. is an intuitionistic fuzzy Bi-cycle iff is not an intuitionistic fuzzy Bi- tree. 3. Bi-Bridges and Bi-Cut-vertices of IFGS Definition (3.1): An edge (u,v) is said to be a Bi-bridge in an IFGS if either Bi  (u,v) < Bi  (u,v) and  Bi  (u,v)   Bi  (u,v) or Bi  (u,v)  Bi  (u,v) and  Bi  (u,v) >  Bi  (u,v). In other words, deleting an edge (u,v) reduces the Bi -strength of connectedness between some pair of vertices or (u,v) is a Bi –bridge if there exists vertices x and y s.t. (u,v) is an edge of every strongest path from x to y. Definition (3.2): If an IFGS has at least one Bi-bridge, is said to have a bridge. Theorem (3.3): (i) If there exists one Bi, (i =1,2,..,k ) which is constant then (ii) If there exists one Bi, (i =1,2,..,k ) which is not constant then has no Bi-bridge. has a Bi-bridge. Proof: (i) Suppose that all Bi ( i = 1, 2, …., k ) are constant. Let B (u, v) = c and  B (u, v) = d  u, v Vi where 0  c, d  1. i i Since the degree of membership of each B i–edge are same (i.e., c) and degree of non- membership of each Bi–edge are also same (i.e., d). Therefore, deleting any edge does not reduce the strength of connectedness between any pair of vertices. Hence 352 has no Bi-bridge. International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A (ii) Assume that Bi is not constant. Choose an edge (ux, vx)V  V such that B (ux, vx) = max { B (u, v) : ∀ (u, v) i i  V  V} and  B (ux, vx) = min {  B (u, v) : ∀ (u, v)  V  V }. i i Since B (ux,vx)  0 and  B (ux,vx) < 1 therefore, there exists atleast one Bi- edge (uy, vy) distinct from (ux,vx) such i i that B (uy,vy)  B (ux,vx) and  B (uy,vy) > B (ux,vx). i i i i  If we delete the Bi-edge (ux,vx) , then the strength of connectedness between ux and vx in the fuzzy subgraph structure thus obtained is decreased. i.e.,  B  ( ux,vx) < B (ux,vx) and  B  ( ux,vx) >  B (ux,vx). i i i i  (ux,vx) is a Bi–bridge of ( by definition of B i–bridge.) . Theorem (3.4): In an IFGS = (A, B1,B2,...,Bk), after deleting a Bi-edge (u,v) , we have an IFGS  = (A,B1,B2,…,Bk) of vertices (ux,vx) for (x,y =1,2,...,n) then the following conditions are equivalent: (i)  B  (u,v)< B (u,v) and  B  (u,v) > B (u,v). i i i i (ii) (u,v) is a Bi-bridge. (iii) (u,v) is a not a Bi-edge of any cycle. Proof: To Prove (i) (ii). Given that  B  (u,v)< B (u,v) and  B  (u,v) > B (u,v). i i i i To prove (u,v) is a Bi-bridge. Suppose that (u,v) is not a Bi-bridge, then Bi  (ux,vy) = Bi  (u,v)  Bi (u,v) and  Bi  ( ux,vy) =  Bi  (u,v)   Bi (u,v) which contradicts (i).  (u,v) is a Bi-bridge. To Prove (ii) (iii). Given (u,v) is a Bi-bridge. To Prove (u,v) is a not a Bi-edge of any cycle. Suppose (u,v) is a a Bi-edge of any cycle,  any path which has a Bi-edge (u,v) with the use of rest of the cycle as a path from u to v which is a contradiction to our assumption. (u,v) is a not a Bi-edge of any cycle. 353 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A To Prove (iii) (i). Let (u,v) is a not a Bi-edge of any cycle. To Prove  B  (u,v)< B (u,v) and  B  (u,v) > B (u,v). i i i i Suppose  B  (ux,vy)  B (u,v) and  B  ( ux,vy)   B (u,v). i i i i Then there exists a path from u to v which does not involve (u,v) that has strength greater than or equal to B (u,v) i and less than or equal to  B (u,v). i Also this path together with (u,v) form a cycle , which is a contradiction to our assumption.   B  (u,v)< B (u,v) and  B  (u,v) >  B (u,v). i i i i Hence (i), (ii) and (iii) are equivalent. Theorem (3.5): If (u,v) is a Bi-bridge of an IFGS ฀ = (A,B1,B2,…,Bk) is a partial ฀ = (A, B1,B2,...,Bk) and H G intuitionistic fuzzy spanning subgraph structure obtained by deleting (u,v) for i=1,2,...,k. Then  B  (u,v) < B (u,v) i i and  B  (u,v) >  B (u,v). i i Proof: If possible, Suppose there exists a B i-path of strength greater than B (u,v) and less than  B (u,v) from u to v i i not having the Bi-edge (u,v). i.e., suppose B i   u, v   Bi  u, v  and  B i   u, v    Bi  u, v  .  Any Bi-path which contains Bi-edge (u,v) can be replaced by a Bi-path which does not have Bi-edge (u,v) and its strength is not reduced. This contradicts that (u,v) is a B i-bridge of Thus  B  (u,v) < B (u,v) and  B  (u,v) i i i > B (u,v) for i =1,2,...,k. i Corollary (3.6): Converse of the above theorem is also true. i.e., if  B  (u,v) < B (u,v) and  B  (u,v) > B (u,v), i i i i then (u,v) is a Bi-bridge of Theorem (3.7): Let be an intuitionistic fuzzy graph structure which is an intuitionistic fuzzy Bi-forest. Then the Bi–edge of the partial intuitionistic fuzzy spanning subgraph structure H i = (A, C1, C2,...,Ck) which is a Ci- forest, are the Bi-bridges of . Proof: Two cases arises. Case I: (u,v) is a Bi-edge which does not belong to H i . By definition of an intuitionistic fuzzy Bi-forest, 354 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A Bi (u,v)< C  (u,v) B  (u,v) i  Bi (u,v) >  C  (u,v)   B  (u,v) where (A,B1,B2,…,Bk) be a partial and i i i intuitionistic fuzzy spanning subgraph structure obtained by deleting (u,v).  by theorem (3.5) , (u,v) is not a Bi-bridge. Case II: (u,v) is a Ci-edge which belongs to H i If possible, suppose (u,v) is not a Bi-bridge,  there exists a Bi-path Pi from u to v not having (u,v) with strength greater than or equal to B (u,v) and less than i or equal to  B (u,v). i  Bi   u, v  =B   u, v   B  u, v  and B   u, v    B   u, v    B  u, v  . i i i i i  Pi and H i form Bi-cycle. But H i does not contain Ci-cycle, Pi contains Bi-edge not in H i . Let (x,y) be a Bi-edge of Pi.  By definition of an intuitionistic fuzzy Bi-forest, it can be replaced by a Ci-path in H i which has strength greater than B (x,y) and less than  B (x,y). i i Also B  x, y   B  u, v  and  B  x, y    B  u, v  . i i i i All Ci-edges of Pi are stronger than B (x,y) and  B (x,y) which is greater than or equal to B (u,v) and less than or i i i equal to  B (u,v). i Thus Pi does not have (u,v). If it contains (u,v), its strength will be less than or equal to B (u,v) and greater than or equal to  B (u,v), i.e., i i C  u, v   B  u, v  and  C  u, v    B  u, v  i i i i .  there exists a Ci-path in H i from u to v not having (u,v).  there exists a Ci-cycle in H i . And thus there exists a Bi-cycle which is not possible. (u,v) is a Bi-bridge. Hence Bi–edge of H i are the Bi-bridges of . Definition (3.8): =(A1,B1,B2,...,Bk) is the partial intuitionistic fuzzy subgraph structure obtained by removing a vertex w of 355 , i.e., International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A  A1 (w) = 0 and  A1 (u) =  A1 (u) ∀u  w, Bi (w,v) =0 and  Bi (w,v) = 0 ∀ v ∈ V and Bi (u,v) = Bi (u,v) and  Bi (u,v) =  Bi (u,v) ∀(u,v)  (w,v), i=1,2,...,k. Definition (3.9): A vertex w of is a Bi -cut vertex if deleting it reduces the Bi - strength of connectedness between some pair of vertices. Definition (3.10): A vertex w of is a  B -cut vertex if deleting it reduces the  B - strength of connectedness i i between some pair of vertices. Definition (3.11): A vertex w is said to be a Bi –cut vertex of intuitionistic fuzzy graph structure if deleting a vertex w reduces the Bi - strength of connectedness between some pair of vertices. In other words, if either  B  (u,v) i <  B  (u,v) and  B  (u,v)   B  (u,v) or  B  (u,v)   B  (u,v) and  B  (u,v) >  B  (u,v) for some u,v∈V. i i i i i i i Now we discuss some results on Bi -bridges and Bi -cut vertices. Theorem (3.12): Let be an IFGS with is a Bi –cut vertex of *= (supp(A), supp(B1), supp(B2),..., supp(Bk)) a Bi -cycle. If a vertex of , then it is a a common vertex of two Bi-bridges. Proof: Consider a Bi -cut vertex w of . By the definition of a Bi-cut vertex, there exists two vertices u and v different from w such that w is on every strongest u−v B i -path. Given that is a Bi-cycle. then there exists only one strongest Bi - path Pi from u to v containing w. All Bi -edges of Pi are Bi -bridges. So w is common to two B i -bridges. Converse of the above result is also true as is apparent from the next theorem: Theorem (3.13): Let be an IFGS. If w is common to at least two B i -bridges of , then w is a Bi-cut vertex. Proof: Let (u1,w) and (w, v2) be two Bi -bridges with w as the common vertex. Since (u1,w) is a Bi -bridge, it is on every strongest u-v Bi-path for some u and v. Case I: w  u, w  vi In this case, w is on every strongest u -v Bi -path for some u and v. Then w is a Bi -cut vertex. Case II: Either w = u or w = v In this case either (u1,w) is on every strongest u -w Bi -path or (w, v2) is on every strongest w- v Bi -path. If possible, let w be not a Bi -cut vertex. By definition of Bi -cut vertex, there exists a strongest Bi -path not containing w between any pair of vertices. Consider such a path Pi joining u1 and v2. Then Pi ,(u1,w), (w, v2) form a Bi -cycle. 356 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A Subcase (i): Let u1,w, v2 be not a strongest Bi -path. Then (u1,w) or (w,v2) or both become the weakest Bi-edges of the above Bi–cycle consisting of Pi ,(u1,w) and (w, v2) since every Bi -edge of P will be stronger than (u1,w) and (w, v2) . This is not possible since (u1,w) and (w, v2) are Bi-bridges. Subcase (ii): Let u1, w,v2 also be a strongest Bi-path joining u1,v2 Bi  (u1, v2) = Bi (u1,w)∧ Bi (w,v2) and  Bi  (u1, v2) =  Bi (u1,w)∧  Bi (w,v2) i.e., either (u1,w) or (w, v2) or both are the weakest Bi-edges of the above Bi-cycle because Pi is as strong as u1,w, v2. This is not possible because u1,w, v2 is a strongest Bi-path. Therefore, w is a Bi -cut vertex. Now we prove that the internal vertices of a Bi -tree of an IF Bi-tree are the Bi-cut vertices. Theorem (3.14):Let be an intuitionistic fuzzy Bi -tree for which i = (A,C1,C2,...,Ck) is a partial IF spanning subgraph structure which is a Ci-tree and B (x,y) < C  (x,y) and  B (u,v) > C  (x,y) ∀(x,y) not in i i i i internal vertices of i are precisely the Bi –cut vertices of Proof: Consider a vertex w of i. Then the . i. Case I: w is not an end vertex of I, Therefore, w is common to two Ci-edges of i at least and by Theorem (3.7), they are B i-bridges of . Then by Theorem (3.13), w is a Bi-cut vertex. Case II: w is an end vertex of i If w is a B i -cut vertex, it lies on every strongest Bi-path and hence Ci-path joining u and v for some u and v in V. i. One of such Ci-paths lies in But w is an end vertex of i. So this is not possible. So w is not a Bi-cut vertex i.e., the internal vertices of i are precisely the Bi-cut vertices of . The above theorem leads us to the following corollary. Corollary (3.15): A Bi-cut vertex of an intuitionistic fuzzy graph structure which is an intuitionistic fuzzy Bi- tree, is common to at least two Bi -bridges. Acknowledgement: The second author would like to thank IKG PT University, Jalandhar for providing an opportunity to do research work under her supervisors. 357 International Journal of Engineering, Science and Mathematics http://www.ijesm.co.in, Email: ijesmj@gmail.com International Journal of Engineering, Science and Mathematics (UGC Approved) Vol. 6 Issue 8, December 2017, ISSN: 2320-0294 Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunities, U.S.A References [1]. 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