Fuzzy and Rough Approximations Operations on Graphs

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1 IOSR Journal of Mathematcs (IOSR-JM) e-issn: , p-issn: Volume 11, Issue 3 Ver. II (May - Jun. 2015), PP Fuzzy and Rough Approxmatons Operatons on raphs M. Shory Department of physcs and Mathematcs, Faculty of Engneerng, Tanta Unversty, Tanta-Egypt, Abstract: In ths paper, we ntroduce a coverng model on connected graph for modfed lower and upper approxmatons operaton on graph vertces wth respect to connected sub graph. We obtan new classes nduced by coverng every sub graphs by lower and upper approxmaton operatons usng fuzzfcaton set of vertces of man graph. Also, propertes of these classes studed and llustrate that wth examples. Keywords: raph theory, Rough set theory, Topology, Fuzzy set theory and Data mnng. I. Introducton Many extensons of rough set theory have been dscussed for approxmatons methods n nformaton systems and other applcatons. Ordnary rough set s a new approach proposed by Pawala [11,18,19,20,21] to deal wth ncomplete or mperfect nformaton. Rough set analyss s based on relaton that descrbes n dstngush ablty of obects. The concepts are represented by ther lower and upper approxmatons. In applcatons, rough sets focus on approxmate reasonng. Rough sets are appled to data mnng, machne learnng, fnance, ndustry, multmeda, medcne, control theory, pattern recognton, and most recently bonformatcs[5,11].rough set s non-statstcal tool for analyss of mperfect Data, data dependences, fuzzy set and decson rules [3,4,5,8, 12,24]. raph theory plays an mportant core for transformaton any engneerng,chemcal, physcal, socal scences, game theores, networs, preferences, and mathematcaton of dscrete spaces.we may fnd new method for classfcaton vertces of graph =(V,E), where V s the set of vertces and E the set of edges, elements of the V depended on the form of ln each vertex n V wth others,ths form can determned by neghborhoods whch classfed V to sets have elements are the same or belong to the same elementary set f they have the same ln value. In [ 17,18 ] neghborhoods of graph vertces used to studed some nequaltes, whch ont the number of each ncdent edge and number of vertces n open sets and closed sets n topologcal structures formed by ths neghborhoods, then appled ths on blood crculaton n the human body and nervous system n some medcal applcaton of neghborhoods of graph vertces. Ths paper may be help us to nows the maxmum and mnmum numbers of vertces used to determned any path n connected graph wth respect to part of ths graph.we ntroduce some mportant new propertes of graph n sense of rough set concepts, we also tred to used classfcaton by neghborhoods of each vertces to obtaned upper and lower sub graph of any part of graph by new mathematcal methods. Ths paper s organzed as follows: In Sec.2 we ntroduced the man concepts of rough set theory. By neghborhoods on connected graph.in Sec.3 we ntroduced new defntons of lower and upper approxmaton operators by membershp functon and precson rough set model II. Rough Set Concepts On Connected raph We try nvestgate rough set concepts on vertces of graph as neghbors[3,8,9,15,17].upper and lower approxmate of connected sub graph vertces wth respect to the man graph determned by neghborhoods, whch cover every sub graph of man graph, lower approxmaton concept mae all vertces and edges les completely n sub graph, upper approxmaton concept mae all vertces and edges possbly les n sub graph. some new rough concepts on ths classes dscussed. Defnton 2.1 Let (V, E)be connected graph, the neghborhood of any vertex s defned as N(v C() ) {v } {v {(N v : v V(), v v ), v V(), v v E}. The coverng graph s defned as E()}. The mnmal neghborhood of any vertex s defned as MN v = {V N v : N v C, v N v }. DOI: / Page

2 Fuzzy And Rough Approxmatons Operatons on raphs Defnton 2.2 Let 1 (V 1, E 1 ) be a connected subgragh of a connected graph (V, E), coverng lower and coverng upper approxmaton operaton are defne on the graph an follows L, U : P(V) P(V) L = {y: y MN v, yv E 1, Vand v V} U = {y: y MN v, yv E 1, Vand v V}. Defnton 2.3 Let 1 (V 1, E 1 ) be a connected subgragh of a connected graph =(V,E). Rough degree of 1 (V 1, E 1 ) wth respect to (V, E),denoted by ρ (),s defned as ρ = 1 L () U 1 () Defnton 2.4 Let (V, E) be asmple connected graph and 1 V 1, E 1, 2 V 2, E 2 are subgraraphs of The postve regon, negatve regon and boundary regon of 2 wth respect to 1 defne as pos 2 = L V 2, neg 2 = V U V 2 and bd 2 = U V 2 L V 2 Defnton 2.5 Let (V, E) be a smple connected graph and 1 V 1, E 1, 2 V 2, E 2 are subgraraphs of. The postve regon, negatve regon and boundary regon of E 2 wth respect to 1 defne as pos E 2 = V 2 K 2 pos 2 and neg E 2 = V 2 K 2 neg 2, Where K 2 = {N V : V V 2 } Example2.1: Consder the Connected graph =(V,E) Fg (2.1) The neghborhoods of vertces are N a = a, b, d, N b = b, c, N c = a, c, d, N d = {b, d} The graphs of the neghborhoods are connected sub graphs of =(V,E). Fg (2.2) DOI: / Page

3 Fuzzy And Rough Approxmatons Operatons on raphs Then the coverng of graph s C = N a, N b, N c, N d and MN a = a, MN b = b, MN c = c, MN d = {d}.consder the sub graph ( V1, E1) Fg (2.3) The set of sub graph s V 1 {a, b,c}. The coverng lower approxmaton of each set of vertces n 1( V1, E ) wth respect to sets of neghborhoods of all vertces of =(V,E)are formed as. L {a} = a, L {b} = {b}, L {c} = c, L {a, c} = a, c, L {a, b} = a, b, L {b, c} = b, c, L V 1 = a, b, c. 1 The coverng upper approxmaton of each set of vertces n ( V1, E1) wth respect to sets of neghborhoods of all vertces of =(V,E)are formed as. U {a} = a, U {b} = {b}, U {c} = c, U {a, c} = a, c, U {a, b} = a, b, U {b, c} = b, c, U V 1 = a, b, c. pos 2 = {b, c, d}, neg 2 = {d} Fg (2.4) We can obtanthe upper and lower approxmaton of any path vertces n 1 V 1, E 1 wth respect to (V, E).Let P1=ae 1 be 2 c be a path form a to c so, the set of vertces of the path s V(P 1 )= {a,b,c}has lower and upper approxmatons as: L(V(P1)) {a, b, c}, U 1 V(P1) {a, b,c}. From above we notce that the pervous of determne can apply the above defnton on undrected connected graph by use the drecton of every edge n two drecton. Example 2.2: Consder the undrected connected graph Fg (2.5) DOI: / Page

4 Fuzzy And Rough Approxmatons Operatons on raphs N a = V, N b = a, b, c, N c = V, N d = {a, c, d} Fg (2.6) C = { N a, N b, N c, N d, MN a = a, c, MN b = a, b, c, MN c = a, c and MN d = {a, c, d} a Fg (2.7) L {a} = L {b} = L {c} =, L {a, c} = a, c, L {a, b} =, L {b, c} =, L V 1 = a, c.u {a} = a, c, U {b} =, U {c} = a, c, U {a, c} = a, c, d, U {a, b} = a, c,u {b, c} = a, c, U V 1 = a, b, c. pos 2 = {a, c},neg 2 = {b} and bd 2 = {d} III. Rough Set Based On Rough Membershp Functon In ths secton we ntroduce the concept of rough set theory by fuzzfcatons crsp set of sub graphs vertces usng ts neghborhoods N (v ) vertces, we obtaned rough membershp functon ndcated the dependence of graph =(V,E) vertces wth sub graphs neghborhoods. The mportant queston be, what s the dfferent between graph operatons (deletons, addng, unons, )and the same operatons on the new classes and parameters. To answer ths queston we must ntroduced some new results. Defnton 3.1 Let =(V,E) and = (V1,E1) be a subgraph. Then the membershp functon of any vertex wth respect to the sub graph 1 s defned as μ v = { N v } V( 1 ), v { N v } N v V() Proposton 3.1 We can redefne the defnton of lower and upper coverng by usng rough membershp functon. Let (V, E) be a smple connected graph and 1 V 1, E 1, 2 V 2, E 2 are subgraraphs of and 2 1 then μ 2 v μ v for all v V() proof From the defnton of membershp and { N v } V( 2 ) { N v } V( 1 ) DOI: / Page

5 Fuzzy And Rough Approxmatons Operatons on raphs We obtaned μ 2 v μ v Proposton3.2 C Let (V, E) be a smple connected graph, 1 V 1, E 1 a subgraraph of and 1 ts complement Wth respect to thenμ v + μ C v = 1 for all v V() proof Proposton 3.3 From the defnton of membershp and N v (V 1 V 1 C ) { N v } Let (V, E) be a smple connected graph and 1 V 1, E 1, 2 V 2, E 2 are subgraraphs of and μ v = 1 ff v L V 2 2 μ v = 0 ff v (V U V 2 ) 3 0 μ v 1 ff v (U V 2 L V 2 ) : obvously From above we can fuzzyfcaton sub graph of a smple connected graph, by usng membershp fucton as characterstc functon of all vertces of a sub graph, we can obtaned a fuzzy set of graph vertces as F v = { v, μ v for every 1 and v V()}. Proposton 3.4 Let (V, E) be a smple connected graph and 1 V 1, E 1, 2 V 2, E 2 are subgraraphs F F F 2 2 F = F F 2 f ether or 2 1 ( ) From μ v 2 max v εv(μ v, μ 2 v ) Pawla forms of rough set theory appled on a specal cases of graph theory n [9,18] Thus, our new wor may be Redefned another forms n the sense of precson coeffcent ( (0.5, 1]). Defnton 3.1 Let (V, E) be a smple connected graph and 1 V 1, E 1 subgraraphs of, -lower and -upper approxmaton of based on rough membershp functon may be defned as L U () = {v : μ v, V 1 } () = {v : μ v 1, V 1 } From above s -exact f and only f L () = U ()Otherwse s -rough. We try to obtan some propertes between the new defntons by comparng rough sub graphs model wth each others Theorem 3.1 Let (V, E) be a smple connected graph, 1 V 1, E 1, subgraraphs of and Y V 1) L =, U = 2) L V = V, U V = V 3) L (L ) = V (U V 4) U (U ) = V (L V 5) L Y L L 6) U Y U U DOI: / Page = 1

6 Fuzzy And Rough Approxmatons Operatons on raphs 7) L Y L L 8) U Y U U 9) If 1 2 then L 2 () L 1 () 10) If 1 2 then U 1 () U 2 () 11) If Y then L L Y 12) If Y then U U Y 13) L U From Defnton 3.1 and membershp propertes Defnton 3.2 Let (V, E) be a smple connected graph, 1 V 1, E 1, subgraraphs of and (0.5,1].For any V - postve regon, -negatve regon and -boundary regon of wth respect to 1 denoted respectvely by pos, neg and bnd () defne as : pos () = {v : μ v, V 1 } neg () = {v : μ v 1, V 1 } bnd () = {v : 1 μ v, V 1 } Proposton 3.5 Let (V, E) be a smple connected graph, 1 V 1, E 1 and 2 V 2, E 2 are sub graphs of, (0.5,1].Te followng satsfes: 1) 2 1 s -exact wth respect to 1 f and only f bnd (V 2 ) =. 2) If 1 and 2 s -exact then 2 s 1 -exact. If 2 and 2 s -exact then 2 s 2 - exact. From above defnton Proposton 3.6 Let (V, E) be a smple connected graph, 1 V 1, E 1 and 2 V 2, E 2 are subgraraphs of, (0.5,1] and 2 1 the followng statements satsfed Obvously 1) pos (V 2 ) = neg (V V 2 ). 2) bnd (V 2 ) = bnd (V V 2 ). 3) If bnd (V 2 ) = then pos (V 2 ) neg (V 2 ) = V. Proposton3.7 Let (V, E) be a smple connected graph, 1 V 1, E 1 and 2 V 2, E 2 are subgraraphs of, (0.5,1] and 2 1 the followng statements satsfed 1) L (V 2 ) L V 2 2) U V 2 U (V 2 ) 3) bnd V 2 bnd V 2 4) neg (V 2 ) neg V 2 Example 3.1 From Example 2.2 μ a = 1, μ b = 2, μ c = 1, μ 3 1 d = 1.Consder ={a,b}, Y={a} L L and U U, L Y L and U Y U 0.6 DOI: / Page

7 Fuzzy And Rough Approxmatons Operatons on raphs IV. Concluson Rough set theory has been regarded as tool to deal wth the uncertanty or mprecson nformaton, the graded rough set model based on two dstnct but related unverses was proposed.but t s stll restrctve for many applcatons, the rough membershp functon based type graded rough set n any graph whch based on coverng neghborhood of all vertces. We have some nterestng propertes and conclusons about rough set model on many sub graphs and paths, whch can help us understand the approxmatons structure of graphs. In the future we wll further study other types of operatons on graphs such that deletons vertces or edges by usng rough sets and so on. References [1]. Buydens,.,. Hac, F.M.,Janssen,P.A.A. Kateman, L.M.C., Postma,.J, A data based approach on analytcal methods ap plyng fuzzy logc n the search strategy and flow charts for the representaton of the retreved analytcal procedures, Chemometr. Intell. Lab, Syst. 25,1994( ). [2]. Collette,T.W., Szladow,A.J.,Use of rough sets and spectral data for buldng predctve models of reacton rate constants, Appl. Spectrosc. 48 Ž1994( ). [3]. Destel,R.,graph theory II, Sprnger-Verage 2005 [4]. Edwards, D.,Zaro,W. olan, R. :An Applcaton of Datalogcr Knowledge Dscovery Tool to Identfy Strong Predctve Rules n Stoc Maret Data, AAAI-93 Worshop. [5]. res, D., Schneder, F.B.,Fundamentals of the new artfcal Intellgence, 2008 ( ). [6]. Hu,.,Zaro, W., Cercone, N., Han, J., RS:A eneralzed Rough Set model, Data Mnng,Rough Sets and ranular Computng, Physcal-Verlag,pp [7]. Inuguch,M. Tanno,T.: eneralzed rough sets and rule, Sprnger-Verlag, Berln, 2002( ). [8]. Inuguch, M., Tanno, T.: On rough sets under generalzed equvalence relatons, Bulletn of Internatonal Rough Set Socety 5(1/2), 2001 ( ). [9]. Kozae,A.M,Abo-Khadra. A., Shory M.,Topology enerated by raphs.jour. of Inst. Of Mathematcs &Computer Scences, Vol. 20, No.1 (2009) [10]. Otto,M. :Fuzzy theory.,a promsng tool for computerzed chemstry, Anal. Chm. Acta 235 Ž1990 ( ). [11]. Pawla,Z.,Rough set theory and ts applcatons, Journal Of Telecommuncatons and Informaton Technology, [12]. Qunlan,J.R.,Ross, J.: R.S. Mchals, et al. ŽEds.., Ma chne Learnng: An Artfcal Intellgence Approach, Toga, Palo Alto, [13]. Shory,M.,Abo-Elnaga,R., Topologcal Propertes on raph VS Medcal Applcaton n Human Heart; Int.J.of Appled Mathematcs,; ISSN: , Vol 15( ). [14]. Shory,M., Abo-Elnaga, R., Concepts of Topologcal Propertes on raph for Treatment the Perpheral Nervous System; Int.J.of Appled Mathematcs ; ISSN: , Vol.28, Issue.1. [15]. SlownsŽEd,R.,Intellgent Decson Support. Handboo of Applcatons and Advances of the Rough Sets Theory, Kluwer Academc Publshers, Dordrecht, [16]. Slowns,R. Stefanows, J. :Rough Classfcaton wth Val ued Closeness Relaton, n: E. Dday, Y. Lechevaller, M. Schader, P. Bertrand, B. BurtschyŽEds.., New Approaches n Classfcaton and Data Analyss, Sprnger, Berln, [17]. Slow ns, R., Vanderpooten, D., A generalzed defnton of rough approxmatons based on smlarty. IEEE Transactons on Data and Knowledge Engneerng 12(2), 2000 ( ). [18]. Stefanows J., On rough set based approaches to nducton of decson rules. In:Rough Sets n Data Mnng and Knowledge Dscovery, Vol 1, Polows L., SowronA., (eds.), PhyscaVerlag, Hedelberg,1998( ).on Knowledge Dscovery n Databases, Washngton, DC, [19]. Yao, Y.Y., Ln, T.Y., eneralzaton of rough sets usng modal logcs. IntellgentAutomaton and Soft Computng 2(2), 1996 ( ). [20]. Yao, Y.Y., Two vews of the theory of rough sets n fnte unverses. InternatonalJournal of Approxmate Reasonng 15, 1996 ( ). [21]. Zaro, W.P.:Rough Sets, Fuzzy Sets and Knowledge Dscovery, Sprnger, New Yor, DOI: / Page