Isomorphism on Intuitionistic Fuzzy Directed Hypergraphs

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1 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 ISSN 50-5 Isomorphsm o Ittostc Fzzy Drected Hypergraphs R.Parath*, S.Thlagaath*,K.T.Ataasso** * Departmet of Mathematcs, Vellalar College for Wome, Eorde-, Tamlad, Ida ** Departmet of Boformatcs ad Mathematcal Modelg, Isttte of Bophyscs ad Bomedcal Egeerg, Blgara Academy of Sceces, sofa, Blgara Abstract- Drected hypergraphs are mch lke stadard drected graphs. Ittostc fzzy drected hypergraphs, lke drected graphs, stadard arcs coect a sgle tal ode to a sgle head ode, hyperarcs coect a set of tal odes to a set of head odes. I ths paper, the somorphsm betwee two ttostc fzzy drected hypergraphs s dscssed. The codto for two ttostc fzzy drected hypergraphs s somorphc also dscssed ad some of ts propertes are also aalyzed. Idex Terms- Ittostc fzzy hypergraph(ifhg), ttostc fzzy drected hypergraph, somorphsm, weak somorphsm, co-weak somorphsm T I. INTRODUCTION here are seeral ways of trodcg the oto of drecto for the edges of a hypergraph. For example, [] a drected hypergraph s obtaed from a hypergraph H, by parttog eery edge of H to two sets of ertces, amely the tal ad the head of the edge. Ths cocept has bee exteded to ttostc fzzy hypergraphs. The athors already trodced the cocept of ttostc fzzy drected hypergraphs[]. Geerally somorphsm betwee two graphs s proed to be a eqalece relato. Kalaa et al., [7] dscssed the propertes of somorphsm ottostc fzzy hypergraphs(ifgs) ad strog IFGs. Radhama et al., trodced the cocept of somorphsm o fzzy hypergraphs ad ther propertes[8]. I ths paper, a attempt has bee made to dere the somorphsm betwee two ttostc fzzy drected hypergraphs. II. PRELIMINARIES I ths secto, some basc deftos relatg to dex matrx represetato of ttostc fzzy graphs (IMIFG)ad ttostc fzzy hypergraphs are ge. I [9], Prof. K.T. Ataasso defed cartesa prodcts of ttostc fzzy sets (IFSs) o dfferet erses. Here, the athors hae defed sx cartesa prodcts of two IFSs oer the same erse. Defto. [5] Let a set E be fxed. Attostc fzzy set (IFS) A E s a obect of the form A {(x, µ A(x), _ A(x)) / x E} where the fcto µ A : E [0,] ad A : E [0,] determe the degree of membershp ad the degree of omembershp of the elemet X E, respectely ad for eery X E, 0 µ (x) (x). Defto. Let X be aersal set ad let V be a IFS oer X the form,, 0 µ ( ) ( ). Sx types of Cartesa prodcts of elemets of V oer X are defed as...,,...,,,,,..., V, A A V V, sch that...,,...,,......,,,..., k k V k k...,,...,,,......,,..., k k V k k...,,...,,m,,...,,max,,...,,,..., V ,,...,,max,,...,,m,,...,,,..., V

2 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 ISSN ,,...,,,,,..., V It mst be oted that t s a IFS, where, t,,,4,5,6 Defto. Attostc fzzy hypergraph H s a ordered par H= V, () V =,,...,, a fte set of ertces () E = E, E,..., E, a famly of ttostc fzzy sbsets of V,, :, 0 () Where E where E ad 0 ( ) ( ), =,,,m ad E V V where : VV [0,] ad : VV [0,] are sch that Ad 0 are the membershp ad omembershp ales of the edge, determed by oe of the Cartesa prodcts t, t,,,4,5,6 for all ad ge defto.. () E, =,,,m () spp E V, =,,,m Here the edges E are IFSs. x ad x ; the ales of ad ca be deote the degree of membershp ad o-membershp of the ertex to edge E. Ths, the elemets of the cdece matrx of IFHG are of the form a,,. The sets (V,E) are crsp sets. Notatos. Hereafter,, that 0., or smply, deotes the degrees of membershp ad omembershp of the ertex V, sch or smply, that 0 deotes the degrees of membershp ad omembershp of the edge, Note If 0, for some ad, the there s o edge betwee ad betwee ad. V V, sch, t s dexed by 0,. Otherwse there exst a edge III. INTUITIONISTIC FUZZY DIRECTED HYPERGRAPHS I ths secto, somorphsm, betwee two ttostc fzzy drected hypergraphs has bee dsscssed. Defto. Attostc fzzy drected hypergraph(ifdhg) H s a par V, E where V s a o empty set of ertces ad E s a set of ttostc fzzy hyperarcs; attostc fzzy hyperarc ee s defed as a par, T e h e, where T e Te, s ts tal, ad hen T e s ts head. A ertex s s sad to be a sorce ertex H f he s ertex d s sad to be a destato ertex H f d T e, for eery e E. Defto. Let E E, E V, wth, for eery e E. A be a hyperarc a IFDHG. The the ertex sets E ad E are called the -set ad the ot-set of the hyperarc E, respectely. The sets E ad E eed ot be dsot. The hyperarc E s sad to be o of the ertces of E ad the ertces of E. Frthermore, the ertces of E are cdet to the hyperarc E ad the ertces of E are cdet from E. The ertces of E are adacet to the ertces of E, ad the ertces of E are adacet from the ertces of E.

3 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 ISSN 50-5 Defto. The order of a IFDHG H V, E Defto.4 The sze of a IFDHG s defed to be S H S H, S H, S H, S H, V, V s defed to be OH O H, O H where O H O H where Defto.5 The -degree of s the mber of hyperarcs that cota ther ot-set, ad s deoted d. Smlarly, the otdegree of s the mber of hyperarcs that cota ther -set,ad s deoted by d H. V H V Defto.6 Cosder the two IFDHGs G V, E ad G V, E.A somorphsm betwee two IFDHGs G ad G, deoted by G G, s a becte map I : V V whch satsfes () I ; I () for eery V, I, I ;, I, I for eery, V Defto.7 A homomorphsm betwee two IFDHGs G V, E ad G V, E, s defed as H : V V s a map whch satsfes () H ; H () for eery V, H, H ;, H, H for eery, V Defto.8 A weak somorphsm betwee two IFDHGs G V, E ad G V, E s defed as I : V V s a becte homomorphsm that satsfes ; I I for eery V Defto.9 A co-weak somorphsm betwee two IFDHGs G V, E ad G V, E s defed as I : V V s a becte homomorphsm that satsfes, I, I ad, I, I for eery, V IV. SOME PROPERTIES OF ISOMORPHISM ON INTUITIONISTIC FUZZY DIRECTED HYPERGRAPHS Theorem 4. For ay two somorphc IFDHG ther order ad sze are same. Proof. If I : G G s a somorphsm betwee the IFDHGs G ad G wth the derlyg sets V ad V respectely, the I ; I for eery V ad, I, I ;, I, I for eery, V We kow that V V O G I O G V V O G I O G,, S H h h S H, V, V,, S H h h S H, V, V Corollary 4. Coerse of the aboe theorem eed ot be tre. Remark 4. If the IFDHGs are weak somorphc the ther order are same. Bt the IFDHGs of same order eed ot be weak somorphc. The followg example llstrates ths. Example 4.4 Let G,,,,, ad G,,,,, be two IFDHGs as ge Fgre ad Fgre respectely

4 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 4 ISSN 50-5 (0.,0.) 5 (0.8,0.) (0.,0.6) 4 (0.4,0.5) 6 (0.4,0.) (0.,0.) Fgre : G 4 (0.,0.) (0.,0.6) 5 (0.4,0.5) (0.8,0.) (0.4,0.) Fgre : G Remark 4.5 If the IFDHGs are co-weak somorphc, the ther szes are same. Bt the IFDHGs of same sze eed ot be co-weak somorphc. Theorem 4.6 If G ad G are somorphc IFDHGs the the degrees of ther ertces are presered. Proof. Let I : V V be a somorphsm of G ad G. By defto.6, we hae, I, I ;, I, I for eery, V d E d I d E d I d E d I d E d I Example 4.7 Cosder the two IFDHGs G ad G whch presere the degree of ertces bt G ad G are ot somorphc.

5 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 5 ISSN 50-5 (0.7,0.) (0.4,0.5) 5 (0.,0.4) 6 (0.5,0.) (,0) 4 (0.6,0.) Fgre : G 4 (0.,0.) (0.7,0.) 6 (0.5,0.4) (0.,0.) (0.4,0.5) Fgre 4: G 5 (0.,0.6) V. ISOMORPHISM ON INTUITIONISTIC FUZZY DIRECTED HYPERGRAPHS USING INDEX MATRIX I ths secto, the dex matrx represetato of two somorphsc ttostc fzzy drected hypergraphs s dscssed. To check somorphsm betwee two ttostc fzzy drected hypergraphs, t s ecessary to check whether () they hae the same mber of ertces, () they hae the same mber of hyperarcs ad () they hae the same mber of ertces wth the same degrees. Cosder the two IFDHGs G,,,, ad G,,,, 4 5 ge Fgre 5 ad Fgre 6 as follows

6 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 6 ISSN 50-5 (0.,0.6) (0.7,0.) (0.5,0.4) 4 (0.,0.7) 5 (0.4,0.5) Fgre 5: G (0.,0.6) 4 (0.,0.7) (0.4,0.5) 5 (0.5,0.4) (0.7,0.) Fgre 6: G The Idex matrx of G s G,, V where V,,,, ad 4 5, , 0, 0, 0.,0.7 0.,0.7 0, 0, 0, 0, 0, 0, 0.,0.7 0, 0, 0, 0, 0.,0.7 0, 0, 0, 0, 0.5,0.4 0, 0, 0, The Idex matrx of G s G,, V where V,,,,, 4 5 ad 4 5 Therefore, the degrees of ertces are calclated ad dsplayed as follows: 4 5 0, 0, 0.,0.7 0.,0.7 0, 0, 0, 0, 0, 0, 0, 0.5,0.4 0, 0, 0, 0, 0.,0.7 0, 0, 0, 0, 0.,0.7 0, 0, 0,

7 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 7 ISSN 50-5 dh dh dh dh 5, we mst hae ether () f ad f 5 or () f 5 ad f dh dh. So f Fally, sce dh 4 dh 5 dh dh 4 () f ad f or () f ad f 4 dh 0, dh ; dh 0, dh, dh 0; dh,0 dh 0, dh ; dh 0, dh 4, dh 4 ; dh 4, dh 5, dh 5 ; dh 5, dh 0, dh ; dh 0, dh, dh 0; dh,0 dh, dh ; dh, dh 4, dh 4 ; dh 4, d 0, d ; d 0, H 5 H 5 H 5. Perhaps ether wll work., we mst hae ether The relabelg s doe by sg () each of the aboe cases to get the map ; 5; ; 4 ; 5 4 permte the rows ad coloms of the dex matrx of G sg ths map to see f we get the dex matrx of G. Otherwse, chage the labels of the graph * G to prodce the graph G * accordg to the aboe permtato ad recalclate the dex matrx. Therefore, the ew dex matrx of G (after labelg of G ) becomes Whch s same as G. Hece, G G, , 0, 0, 0.,0.7 0, 0, 0, 0, 0, 0, 0, 0.,0.7 0, 0, 0, 0, 0.,0.7 0, 0, 0, 0, 0.5,0.4 0, 0, 0, VI. CONCLUSION A graph somorphsm search s a mportat problem of graph theory. It prodes a becte correspodece whch presere adacet relato betwee ertex sets of two graphs. I ths paper, a attempt has bee made to defe a somorphsm betwee two ttostc fzzy hypergraphs. For, sx types of cartesa prodcts of two IFSs oer the same erse are defed. Also ttostc fzzy drected hypergraph s defed sg cartesa prodcts, addto to the dex matrx represetato of somorphsm betwee two IFDHGs. ACKNOWLEDGMENT The athors R. Parath ad K.T.Ataasso wold lke to thak the Departmet of Scece ad Techology, NewDelh, Ida ad Mstry of Edcato ad Scece, Sofa, Blgara, for ther facal spport to the Blateral Scetfc Cooperato Research Programme INT/Blgara/B-/08 ad BI REFERENCES [] A.Rosefeld, Fzzy graphs, Fzzy sets ad ther applcatos (L.A. Zadeh, K.S. F, M.Shmra,Eds),Academc press, New York, 975, pp [] G.Gallo, G.Logo, S.Ngye, S.Pallotto, Drected hypergraphs ad applcatos, Dscrete Appled Mathematcs, 40, 99, pp

8 Iteratoal Joral of Scetfc ad Research Pblcatos, Volme, Isse, March 0 8 ISSN 50-5 [] K.T. Ataasso ad Athoy Shao, A frst step to a theory of the ttostc fzzy graphs, Proceedgs of the frst workshop o Fzzy Based Expert systems (D Lako,Ed.), Sofa 8-0, September 994, pp [4] K.T. Ataasso ad Athy Shao, Ittostc fzzy graphs from, ad, -leels, Notes o Ittostc Fzzy Sets, 995, pp. -5. [5] K.T.Ataasso, Odex matrx represetato of ttostc fzzy graphs, Notes ottostc fzzy sets, 4, 00, pp [6] R.Parath ad M.G. Karambga, Ittostc fzzy graphs, Proceedgs of 9 th Fzzy Days Iteratoal coferece o Comptatoal Itellgece, Adaces soft comptg: comptatoal tellgece, Theory ad applcatos, Sprger- Verlag, 0, 006, pp [7] O.K. Kalaa, R.Parath ad M.G. Karambga, A stdy o Ataasso s ttostc fzzy graphs, Iteratoal Coferece o Fzzy Systems (FUZZ-IEEE 0) Jly 0, pp [8] C.Radhama ad C. Radhka, Isomorphsm o fzzy hypergraphs, IOSR Joral of Mathematcs,, 0, pp. 4-. [9] K.Ataasso, Ittostc fzzy sets: Theory ad Applcatos, Physca- Verlag, New York, 999. [0] J. N. Mordeso ad P. S. Nar,Fzzy graphs ad fzzy hypergraphs, Physca erlag, Hedelberg 998; Secod Edto 00. [] R.Parath, S.Thlagaath ad K.T.Ataasso, Ittostc fzzy drected hypergraphs, sbmtted. AUTHORS Frst Athor R.Parath, Ph.D., Departmet of Mathematcs, Vellalar College for Wome, Erode-, Tamlad, Ida. paaraths@redffmal.com. Secod Athor S.Thlagath, M.Phl., Vellalar College for Wome, Erode-, Tamlad, Ida. ad emal address thlak.pr@gmal.com. Thrd Athor K.T.Ataasso, D.Sc., Departmet of Boformatcs ad Mathematcal Modelg, Isttte of Bophyscs ad Bomedcal Egeerg, Blgara Academy of Sceces, sofa, Blgara. krat@bas.bg. Correspodece Athor S.Thlagaath, thlak.pr@gmal.com, cotact mber