Global Journal of Pure and Appled Mathematcs. ISSN 0973-768 Volume 3, Number 0 07), pp. 7365-7376 Research Inda Publcatons http://www.rpublcaton.com Double Layered Fuzzy Planar Graph J. Jon Arockaraj Assstant Professor & Head of the Dept.,PG & Research Department of Mathematcs,St. Joseph s College of Arts and Scence Autonomous) Cuddalore, TamlNadu, Inda. B.Rejna Research Scholar,PG & Research Department of Mathematcs, St. Joseph s College of Arts and Scence Autonomous), Cuddalore,TamlNadu, Inda. Abstract Fuzzy planar graph s a very mportant subclass of fuzzy graph. In ths paper, we defne a new fuzzy graph double layered fuzzy planar graph FPG and weak double layered fuzzy planar graph. And we dscuss the basc propertes of t. Keywords: Fuzzy graphs, fuzzy planar graph, double layered fuzzy planar graph, weak double layered fuzzy planar graph, fuzzy faces.. INTRODUCTION Fuzzy set theory was ntroduced by Zadeh n 965[0]. Fuzzy graph theory was ntroduced by Azrel Rosenfeld n 975[]. Though ntroduced recently, t has been growng fast and has numerous applcatons n varous felds. Durng the same tme Yeh and Bang[] have also ntroduced varous concepts n connectedness wth fuzzy graph. Nagoorgan and Malarvzh[3] have defned dfferent types of fuzzy graphs and dscussed ts relatonshps wth somersm n fuzzy graphs. Abdul-jabbar and Naoom[8] ntroduced the concept of fuzzy planar graph. Also, Nrmala anddhanabal [7] defned specal fuzzy planar graph. A.Pal, S.Samanta and M.Pal[4] have defned
7366 J. Jon Arockaraj and B.Rejna fuzzy planar graph n a dfferent concept where crossng of edge are allowed. Pathnathan and JesnthaRoslne[6] have ntroduced double layered fuzzy graph. The Intutonstc double layered fuzzy graph s gven by JesnthaRoselne and Pathnathan[9]. In ths paper we defne Double Layered Fuzzy Planar Graph FPG and we dscuss some propertes. Secton two contans the basc defntons n fuzzy graph and fuzzy multgraph, n secton three we ntroduce a new fuzzy graph called double layered fuzzy planar graph, secton four presents the theoretcal concept of double layered fuzzy planar graph and fnally we gve concluson on FPG.. PRELIMINARIES A graph can be drawn n many dfferent ways. A graph may or may not be drawn on a plane wthout crossng of edges. A drawng of a geometrc representaton of graph on any surface such that no edges ntersect s called embeddng []. A graph G s planar f t can be drawn n the plane wth ts edges only ntersectng at vertces of G. So the graph s non planar f t cannot be drawn wthout crossng. Several defntons of strong edge are avalable n lterature. Among them the defnton of [] s more sutable for our purpose. The defnton s gven below: For the fuzzy graph V,, ) an edge x, y) s called strong [] f mn a), a, and weak otherwse. A multlaton s a graph that may contan multple edges between any two vertces, but t does not contan any self loops. Defnton. []A fuzzy graph G V,, ) s a non-empty set V together wth a par of functons : V [0, ] and : V V [0, ] such that for all a, b V, a, a), where a), and a, represent the membershp values of the vertex a and of the edge a, n G respectvely. Defnton. [4]LetV be a non-empty set and : V [0, ] be a mappng. Also let b V V V,, where max a, 0 E a,, a, ),,,... P a, ) be a fuzzy multset of V such for all,...pab that a, a) ab Then V,, E) s denoted as fuzzy multgraph where a) and p. ab a b, ) represent the membershp value of the vertex a and the membershp value of the edge a, n respectvely.
Double Layered Fuzzy Planar Graph 7367 Defnton.3 [4]Let be a fuzzy multgraph and for a certan geometrcal representaton P, P,... PN be the pont of ntersectons between the edges s sad to be fuzzy planar graph wth fuzzy planarty value f, where f = { I I... I P P PN } It s obvous that f s bounded and the range of f s 0< f. 3. DOUBLE LAYERED FUZZY PLANAR GRAPHFPG) Defnton 3..Let V,, E) be a fuzzy multgraph wth the underlyng crsp multgraph V,, E ).The vertex set of ) be E. The geometrcal representaton P,...,, P PN be the ponts of ntersectons between the edges ) V,, E ) s sad to be double layered fuzzy planar graph FPG) wth double layered fuzzy planarty value f = { I P I P... I P N. } f, where It s obvous that f s bounded and the range of f s 0 < f. Algorthm for double layered fuzzy planar graph: Step:We consder the fuzzy planar graph. Step: We choose the edges of the fuzzy planar graph to consder new vertex of the fuzzy planar graph.we get new vertex set of the double layer fuzzy planar graph V E). Step3:In the gven graph, draw edges for the adjacent vertex and also for the adjacent edges. Step4:Here we wll get a new graph whch has many ntersectng edges. Plot the ntersectng pont as P P,..., P, N. Step5:Evaluate the value for the ntersectng pont usng the formula I a, a, and I a) ) P k I a, I c, d ) where k,,..., N Step6: If the double layered planarty value s 0 < f then the graph s double layered fuzzy planar graph.
7368 J. Jon Arockaraj and B.Rejna Remark: 3...We only consder mnmal ntersectng ponts of the double layered fuzzy planar graph. Example:3...Consder the fuzzy planar graph, whose crsp graph wth n=4 vertces. s a cycle Fgure: A Fuzzy Planar graph V,, E) Fgure: Double layered fuzzy planar graph ) V,, E )
Double Layered Fuzzy Planar Graph 7369 Here we calculate the ntersectng value at the ntersectng pont between two edges.two edges b, c) and e, ) are ntersected where a ) =0.7), =0.8), e 5 E e ) =0.45), E e ) =0.4), E b, c) =0.5, E e ) E ) =0.3see fgure ). Strength 5 e5 0.45 of the edge b, c) s =0.57.e.) I b, c) =0.64 and that of, 5) 0.7 e e s 0.3 0. 4 I =0.75. Thus the ntersectng value at the pont s e, e5 ) 0.65 0.75 P 0.93 P 0.83 3 P 0.83 4 P 0.7 5 P 0.79 6 =0.695. P 0. 695. Smlarly, we can fnd = 0.75.e.) Consder the fuzzy planar graph wth n=5 vertces. Fgure3: A Fuzzy Planar graph V,, E)
7370 J. Jon Arockaraj and B.Rejna Fgure4: Double layered fuzzy planar graph ) V,, E ) Consder the fuzzy planar graph wth n=6 vertces. Fgure5: A Fuzzy Planar graph V,, E)
Double Layered Fuzzy Planar Graph 737 Fgure6: Double layered fuzzy planar graph ) V,, E ) Defnton:3..A double layered fuzzy planar graph ) s called weak double layered fuzzy planar graph f the double layered fuzzy planarty value of the graph s less than or equal to 0.5. Defnton: 3.3. Let ) V,, E ) be a double layered fuzzy planar graph and E a,, a,,,,..., P a V V P max{ a, 0}. A ab fuzzy face of ) s a regon bounded by the set of fuzzy edges E E of a geometrc representaton of ). The membershp value of the fuzzy face s a, mn,,,..., P { a) ab ab a, E
737 J. Jon Arockaraj and B.Rejna A fuzzy face s called weak fuzzy face f ts membershp value s < 0.5 and otherwse strong face. Every double layered fuzzy planar graph has an nfnte regon whch s called outer fuzzy face. Other faces are called nner fuzzy faces. 4. THEORETICAL CONCEPTS Theorem: 4.. Let ) be a weak double layered fuzzy planar graph. The number of pont of ntersectons between strong edges n ) s P, P,..., PN. Proof: Let ) V,, E ) be a weak double layered fuzzy planar graph. Let, f possble, ) has one pont of ntersectons between two strong edges n ). For any strong edge V, V ), V, V ) ), V, V ) { V ) V )}. So I V, ) 0.5 V j Thus for two ntersectng strong edges V, V ), V, V ) ), and V3, ), V3, ) ), I I V, V ) V3, ) Therefore 0.5. That s f = > 0.5. I P I P 0.5, Then +0.5.5. It contradcts the fact that the fuzzy graph s a weak double layered fuzzy planar graph. So number of pont of ntersectons between strong edges cannot be one. It s clear that f the number of pont of ntersecton of strong edges n one then the double layered fuzzy planarty value f 0.5. Smlarly, f the number of pont of ntersectons of strong edges ncreases, the double layered fuzzy planarty value decreases. Any FPG wthout any crossng between edges s a strong FPG. But FPG contans large number of ntersectng ponts. So, the FPG s a weak double layered fuzzy planar graph. Thus we conclude that the number of pont of ntersectons between strong edges n ) s P, P,..., PN. Theorem: 4..Let ) be a double layered fuzzy planar graph. Then ) s not strong double layered fuzzy planar graph.
Double Layered Fuzzy Planar Graph 7373 Proof:Let ) V,, E ) be a weak double layered fuzzy planar graph. Let, f possble, ) has at least two ntersectng ponts P and P. For any strong edge V, V ), V, V ) ), V, V ) { V ) V )}. So I V, ) V 0.5 j Thus for two ntersectng strong edges V, V ), V, V ) ), and V3, ), V3, ) ) I I V, V ) V3, ) Then I I. P P Therefore f = 0.5. That s I P I P I P not strong double layered fuzzy planar graph. 0.5, smlarly I P 0.5 0.5. It s clearly that double layered fuzzy planar s Theorem:4.3. Let ) be a double layered fuzzy planar graph wth double layered planarty value 0< f. Then ) has strong and weak fuzzy faces. Proof:Let ) be a double layered fuzzy planar graph. Let F and F be two fuzzy faces, j F s bounded by the edges, V, V ), V, V ) ), V3, ), V3, ) ) and k, V5 ),, V5 ) ) Then, j k V, V ) V3, ) V5, V6 ) Mn,, >0.5.e.) ts membershp value V ) V ) V3 ) ) V5 ) V6 ) s 0.5. Smlarly, F s fuzzy bounded face wth membershp value 0.33. Every strong fuzzy face has membershp value s greater than 0.5. So, every edge of a strong fuzzy face s a strong fuzzy edge, and every weak fuzzy face has membershp value s less than 0.5. So, ) has strong and weak fuzzy faces.
7374 J. Jon Arockaraj and B.Rejna Example:4.3.. In fgure 7, F, F,..., Fare twenty one fuzzy faces. Fgure7: Double layered fuzzy planar graph ) V,, E ) F s bounded by the edges e,,0.4), b, e),0.3), e, e),0.35) the membershp value of the fuzzy face s, a, mn,,..., p a) membershp value s 0.67. ab a, E..e.) 0.8,0.67,0.78 0. 67 mn, ts So, F s a strong fuzzy face. Smlarly, we can fnd F, F3,..., F.Here F0s a fuzzy bounded face and Fs the outer fuzzy face wth membershp value 0.33 obvously, F0and Fare weak fuzzy faces.
Double Layered Fuzzy Planar Graph 7375 CONCLUSION Ths study descrbes the double layered fuzzy planar graph and ts propertes. We have defned a new term called double layered fuzzy planarty value of a fuzzy graph. If the double layered fuzzy planarty value of a fuzzy graph s one then no edge crosses other edge. Ths s a very mportant concept of fuzzy graph theory. Double layered fuzzy planarty value s less than 0.5 because ths graph contan large number of ntersectng ponts. So double layered fuzzy planar graph s defned as weak double layered fuzzy planar graph. REFERENCES [] A.Rosenfeld, Fuzzy graphs, In: L.A.Zadeh, K.S.Fu, K.Tanaka and M.Shmura, Edtors), Fuzzy sets and ts applcaton to cogntve and decson process, Academc press, New York, 975, 77-95. [] R.T.Yeh and S.Y.Bang, Fuzzy relatons,fuzzy graphs and ther applcatons to clusterng analyss, n: L.A.Zadeh, K.S.Fu, K.Tanaka and M.Shmura, Edtors), Fuzzy sets and ts applcaton to cogntve and decson process, Academc press, New York,975,5-49. [3] A.Nagoorgan and J.Malarvzh, Propertes of μ-complement of a fuzzy graph, Internatonal Journal of Algorthms, Computng and Mathematcs, 3) 009, 73-83. [4] Sovansamanta, Anta Pal, and Madhumangal Pal, New concepts of fuzzy planar graphs Internatonal journal of Advanced Research n Artfcal Intellgence, vol.3,no.,04,pp.5-59. [5] Sovansamanta, and Madhumangal Pal, Fuzzy planar graphs [6] T.Pathnathan and J.JesnthaRoslne Double layered fuzzy graph Annals of Pure and Appled Mathematcs, vol.8, no., 04, pp.35-43. [7] G.Nrmala and K.Dhanabal, Specal planar fuzzy graph confguratons, Internatonal journal of Scentfcand ResearchPublcatons, vol., no.7, 0, pp.-4. [8] N.Abdul-jabbar and J.H.Naoom and E.H.Ouda, Fuzzy dual graph, Journal of Al-Nahran Unversty, vol., no.4, 009, pp.68-7. [9] T.Pathnathan and J.JesnthaRoslne Intutonstc double layered fuzzy graph ARPN Journal of Engneerng and Appled Scence, vol.0, no., 05, pp.543-547. [0] L.A.Zadeh, Fuzzy sets, Informaton and Control, 965, pp.338-353.
7376 J. Jon Arockaraj and B.Rejna [] C. Eslahch and B. N. Onaghe, Vertex strength of fuzzy graphs, Internatonal Journal of Mathematcs and Mathematcal Scences, Volume 006, Artcle ID 4364, Pages -9, DOI 0.55/IJMMS/006/4364. [] V.K.Balakrshnan, Graph Theory, McGraw-Hll,997.