Itratoal Joural of Mathmatal Arhv-6), 0, 07- Avalabl ol through wwwjmafo ISSN 9 06 ON THE LINE-CUT TRANSFORMATION RAPHS B BASAVANAOUD*, VEENA R DESAI Dartmt of Mathmats, Karatak Uvrsty, Dharwad - 80 003, Ida Rvd O: 06-0-; Rvsd & Atd O: 30-0-) ABSTRACT I ths ar, w trodu l-ut trasformato grahs W vstgat som bas rorts suh as ordr, sz, otdss, grah quatos damtrs of th l-ut trasformato grahs 00 Mathmats Subjt Classfato: 0C Kywords: utot, l grah, l-ut trasformato grahs INTRODUCTION By a grah = V, E) V ), E ), W ) ), w ma a sml, ft, udrtd grahs wthout solatd ots For ay grah, lt U dot th ot st, l st, utot st blok st of, rstvly Th ls utots of a grah ar alld ts mmbrs Etrty of a ot u V ) s dfd as u) = max{ d u, v) : v V )}, whr d u, v) s th dsta btw u v Th mmum maxmum trts ar th radus r ) damtr dam ) of, rstvly A utot of a otd grah s th o whos rmoval rass th umbr of omots A osarabl grah s otd, otrval has o utots A blok of a grah s a maxmal osarabl subgrah A blok s alld dblok of a grah f t otas xatly o utot of Th l grah L ) of s th grah whos ot st s E ) whh two ots ar adjat f oly f thy ar adjat Th jum grah J ) of s th grah whos ot st s E ) whh two ots ar adjat f oly f thy ar oadjat [] For grah thort trmology, w rfr to [, 7] LINE-CUT TRANSFORMATION RAPHS Isrd by th dfto of total trasformato grahs [0] blok-trasformato grahs [3], w trodu th grah valud futos amly l-ut trasformato grahs w df as follows Dfto: Lt = V, E) b a grah, lt α, β b two lmts of E ) W ) W say that th assoatvty of α β s + f thy ar adjat or dt, othrws s Lt b a -rmutato of th st { +, } W say that α β orrsod to th frst trm x of f both α β ar E ) α β orrsod to th sod trm y of f o of α β s E ) th othr s W ) Th l-ut trasformato grah of s dfd o th ot st E ) W ) Two ots α β of ar jod by a l f oly f ths assoatvty s osstt wth orrsodg trm of S thr ar four dstt -rmutatos of { +, }, w obta four l-ut trasformatos of amly + +, +, + Corrsodg Author: B Basavaagoud* Itratoal Joural of Mathmatal Arhv- 6), May 0 07
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs / IJMA- 6), May-0 I othr words, lt b a grah, x, y b two varabls takg valus + or Th l-ut trasformato grah s th grah havg E ) W ) as th ot st, for α, β E ) W ), α β ar adjat f oly f o of th followg holds: ) α, β E) α β ar adjat f x = + ; α β ar oadjat f x = ) α E), β W ) α β ar dt f y = + ; α β ar odt f y = It s trstg to s that + + s xatly th lt grah of [6] It s also alld as l-ut grah of [] May ars ar dvotd to lt grah [,, 6, 8] Th ot vrtx ) of orrsodg to a utot l ) of s rfrrd to as utot l) A grah all ts four l-ut trasformato grahs ar show Fg I l-ut trasformato grahs th l vrts ar dotd by dark rls th utot vrts ar dotd by lght rls Th followg wll b usful th roof of our rsults Rmark: L ) s a dud subgrah of + + + Rmark: J ) s a dud subgrah of + Thorm: [] If s otd, th L ) s otd Thorm: [] Lt b a grah of sz q Th J ) s otd f oly f otas o l that s adjat to vry othr ls of ulss = K or C Thorm: 3 [6] A otd grah s somorh to ts + + f oly f s a yl 0, IJMA All Rghts Rsrvd 08
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs Th followg thorm dtrms th ordr sz of a l-ut trasformato grahs / IJMA- 6), May-0 Thorm: Lt b a otrval otd, q) -grah wth ot st V ) = { v, v,, v }, l st E) = {,,, q}, utot st W ) = {,,, m} blok st U ) = { B, B,, B}, th ots of hav dgr blogs C B ) b th umbr of d L b th umbr of ls to whh utot utots of a otd grah whh ar th ots of th blok q + + C B ) ) = Th sz of + = q + q + Th sz of + = 3 Th sz of = m d + q L ) = = m d + L = = m d = = q + + q L ) B Th th ordr of Proof: If s a otd grah wth ots q ls, th L ) has q ots Lt B ) umbr of utots of a otd grah whh ar th ots of th blok + utot grah C ) s gv by C B ) ) S L ) ) Thrfor th ordr of = + = q + C B ) ) = s C b th B Th th umbr of ots th J hav sam umbr of ots Th umbr of ls + s th sum of th umbr of ls L ) sum of th umbr of ls odt wth th utots Thus th sz of + = q + m d + q L ) = = Th umbr of ls + s th sum of th umbr of ls J ) sum of th umbr of ls dt wth th utots Thus th sz of + = q + = d m + L = 3 Th umbr of ls s th sum of th umbr of ls J ) sum of th umbr of ls odt wth th utots Thus th sz of = 3 CONNECTEDNESS OF Th frst thorm s wll-kow q + = d m + q L ) Thorm: 3 For a gv grah, + + s otd f oly f s otd Thorm: 3 For ay grah wth q ) K, ) K K,, r =, + s otd f oly f 0, IJMA All Rghts Rsrvd 09
) B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs B = v) K B ), = / IJMA- 6), May-0 Proof: Suos a grah satsfs odtos ), ), ) v) W rov th rsult by followg ass Cas- If s otd, th w hav th followg subass Subas-: If s a blok, th larly + = L ) s otd Subas-: If has at last o utot, th L ) s otd subgrah of + also ah utot vrtx s adjat to at last o l vrtx baus vry utot s odt wth at last o l H + s otd Cas-: If s dsotd wth,,, omot s ot a star wth at last o utot For vry ar of l vrtx orrsodg ls omots By odtos ), ) v) o of th j rstvly ar o adjat ar otd by utot vrtx vry ar of l vrtx x y whos orrsodg ls x adjat + Thrfor + s otd Th ovrs s obvous j whos for y rstvly ar adjat ar Thorm: 33 For ay grah wth q, + s otd f oly f K, C, K K x whr x 3, s ay l K ) has o l whh s adjat to all othr ls s odt to a utot Proof: lt b a otd grah wth q, K3, C, K, K x whr x s ay l K ) has o l whh s adjat to all othr ls s odt to a utot Th to rov + s otd W osdr th followg ass Cas If s otd th w hav th followg subass Subas-: If s blok s otd K, C, K K x whr x s ay l K ), th larly + = J ) 3, Subas-: If has atlast o utot th w hav th followg subsubass Subsubas-: If otas o l whh s adjat to all othr ls, th by Thorm, J ) s otd subgrah of, h + s otd Subsubas-: If otas at last o l whh s adjat to all othr ls, larly s dt wth a utot, th l vrts utot vrts ar otd + Thrfor + s otd Cas-: If s ot otd th J ) s otd subgrah of + ah utot vrtx s adjat to atlast o l vrtx baus vry utot s dt wth atlast o l H + s otd Covrsly, larly + s otd for ay grah of sz q, K, C, K K x whr x s ay 3, l K ) has o l whh s adjat to all othr ls s odt to a utot Thorm: 3 For ay grah wth q, s otd f oly f K,, K3, C, K, K x whr x s ay l K ) has o l whh s adjat to all othr ls s dt to a utot 0, IJMA All Rghts Rsrvd 0
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs / IJMA- 6), May-0 Proof: lt b a otd grah wth q, K,, K3, C, K, K x whr x s ay l K ) has o l whh s adjat to all othr ls s dt to a utot Th to rov s otd W osdr th followg ass Cas- If s otd th w hav th followg subass Subas-: If s blok = ) J s otd K,, K3, C, K, K x whr x s ay l K ), th larly Subas-: If has atlast o utot th w hav th followg subsubass Subsubas-: If otas o l whh s adjat to all othr ls, th by Thorm, J ) s otd subgrah of H s otd Subsubas-: If otas at last o l whh s adjat to all othr ls, s K, thrfor thr s atlast o l whh s odt wth utot, th l vrts utot vrts ar otd Thrfor s otd Cas-: If s ot otd, th J ) s otd subgrah of ah utot vrtx s adjat to atlast o l vrtx baus vry utot s odt wth atlast o l H s otd Covrsly, larly s otd for ay grah of sz q, K,, K3, C, K, K x whr x s ay l K ) has o l whh s adjat to all othr ls s dt to a utot RAPH EQUATIONS AND ITERATIONS OF For a gv grah orator Φ, whh grah s fxd udr th orator Φ?, that s for a otd grah, L ) f oly f s a yl [9] For a gv l-ut trasformato grah ) = ) ) = [ ] for Th somorhsm of + + s show [6], w df th trato of Φ ) as follows: Thorm: Lt b a otd grah Th L ) = + f oly f s a blok? It was kow that Proof: Suos s a blok It s kow that has o utots Th + has q ots By dfto of L ) t has q ots Clarly L ) = + Covrsly, suos L ) = + Assum s ot a blok Th thr xst at last o utot It s kow that L ) has q ots whr as th umbr of ots of + ar th sum of th umbr of ls utots of L has lss umbr of ots tha + Clarly + L), a otradto Thus ) Thorm: A otd grah s somorh to ts + f oly f s a yl Proof: W kow that a otd grah s somorh to ts l grah f oly f t s a yl Also from Thorm, L ) = + f oly f s a blok Thrfor a otd grah s somorh to ts + f oly f s a yl + ) Corollary: 3 For a otrval otd grah, = f oly f s a yl 0, IJMA All Rghts Rsrvd
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs / IJMA- 6), May-0 Thorm: Lt b a otd grah Th J ) = + f oly f s a blok Proof: Suos s a blok It s kow that has o utots Th + has q ots By dfto of J ) t has q ots Clarly J ) = + Covrsly, suos J ) = + Assum s ot a blok Th thr xst at last o utot It s kow that J ) has q ots whr as th umbr of ots of + ar th sum of th umbr of ls utots of J has lss umbr of ots tha + Clarly + J ), a otradto Thus ) Thorm: A otd grah s somorh to ts + f oly f s K, or C Proof: Suos + = Assum, K, W osdr th followg ass C Cas-: Suos s a blok If C, th + J ), a otradto Cas-: Suos s ot blok If K,, th thr xsts atlast o l whh s odt wth utot Thrfor +, a otradto Covrsly, f s K, or C, th larly + = Thrfor a otd grah s somorh to ts + f oly f s K, or C Corollary: 6 For a otrval otd grah, = +) f oly f s K, or C Thorm: 7 Lt b a otd grah Th = J ) f oly f s a blok Proof: Suos s a blok It s kow that has o utots Th has q ots By dfto of J ) t has q ots Clarly J ) = Covrsly, suos J ) = Assum s ot a blok Th thr xst at last o utot It s kow that J ) has q ots whr as th umbr of ots of ar th sum of th umbr of ls utots of J has lss umbr of ots tha Clarly J ), a otradto Thus ) Thorm: 8 A otd grah s somorh to ts f oly f s C Proof: W kow that a otd grah s somorh to ts jum grah f oly f t s C Also from Thorm 7, J ) = f oly f s a blok Thrfor a otd grah s somorh to ts f oly f s C Corollary: 9 For a otrval otd grah, = ) f oly f s C DIAMETERS OF Thorm: For ay otrval otd grah suh that + + s otd, dam + + ) dam ) + Proof: Lt b a otd grah W osdr th followg thr ass Cas-: Assum s a tr, th larly dam + + )<dam)+ 0, IJMA All Rghts Rsrvd
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs / IJMA- 6), May-0 Cas-: Assum s a yl dam ) + C, 3, th from Thorm 3, + + = L ) Thrfor dam + + ) Cas-3: Assum otas a yl C, 3 orrsodg to a yl C, L C ) s a subgrah + + Thrfor dam + + ) dam ) + From all th abov ass, dam + + ) dam ) + Thorm: For ay otrval otd grah wth atlast o utot otd, dam + ) s atmost suh that + s K, Proof: Lt b a otrval otd grah wth atlast o utot W osdr th followg ass K,, suh that + s otd Cas-: Lt b l vrts of + If th ls ar adjat th dd +, ) = If th ls ar oadjat th thr xsts a l adjat to both th ls or thr xsts a utot odt wth both th ls I both th ass dd +, ), so that, th dsta btw ay two l vrts + s atmost Cas-: Lt b utot vrts of + W osdr th followg subass Subas-: If th utots ar oadjat s a l odt wth both, th ) s a ath of lgth +, h dd +, ) = Subas-: If th utots ar oadjat s a l dt wth but odt wth, th ar otd by a ath of lgth +, h dd +, ) = Subas-3: Lt b adjat If all th ls of ar dt wth, th dd +, ) = KK 3 KK 3 If ar adjat thr xsts a l whh s odt wth, th th utot vrts ar otd by l vrtx +, h dd +, ) = I all th ass th dsta btw ay two utot vrts + s atmost, th dd +, ) = Cas-3: Lt a l dd +, ) b utot vrtx l vrtx rstvly of + If th utot s odt wth, th If th utot s dt wth a l = aa llllllll 3 aa llllllll Thrfor th dsta btw utot vrtx l vrtx + s atmost H from all th abov ass, dam + ) s atmost Thorm: 3 For ay grah of sz q, K3, C, K, K x whr x s ay l K ) has o l whh s adjat to all othr ls s odt to utot suh that + s otd, dam + ) s atmost 3 Proof: Lt b a grah of sz q, K, C, K K x whr x s ay l 3, K ) has o l whh s adjat to all othr ls s odt to utot suh that + s otd W osdr th followg ass 0, IJMA All Rghts Rsrvd 3
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs / IJMA- 6), May-0 b th l vrts of + If th ls ar oadjat, th Cas-: Lt dd +, ) = If th ls ar adjat, th thr xsts a l whh s oadjat to both th ls or thr xsts a utot dt to both th ls, th dd +, ) = Othrws dd +, ) = 3 Thrfor dd +, ) 3, so that th dsta btw ay two l vrts + s atmost 3 Cas-: Lt b utot vrts + W osdr th followg subass Subas-: If th utots ar adjat, th th utot vrts otd by a l vrtx h dd +, ) = orrsodg to a l whh s dt wth both utots + ar, Subas-: If th utots ar oadjat thr xsts ls suh that a l s dt wth a utot a l s dt wth a utot, th ar otd by a ath of lgth 3 +, h dd +, ) = 3 I both subass th dsta btw utot vrts + s atmost 3 Cas-3: Lt b l vrtx utot vrtx rstvly of + If a l s dt wth utot, th dd +, ) = If a l s odt wth utot thr xst a l whh s dt wth utot oadjat to a l, th lgth +, h dd +, ) = H from all th abov ass, dam + ) s atmost 3 Thorm: For ay grah ar otd by a ath of of sz q, K K C K K x,, 3,,, whr x s ay l K ) has o l whh s adjat to all othr ls s dt to utot suh that s otd, dam ) s atmost Proof: Lt b a grah of sz q, K,, K3, C, K, K x whr x s ay l K ) has o l whh s adjat to all othr ls s dt to utot suh that s otd W osdr th followg ass b th l vrts of If th ls ar oadjat, th Cas-: Lt dd, ) = If th ls ar adjat, th thr xsts a l whh s oadjat to both th ls or thr xsts a utot odt to both th ls, th dd, ) = If thr s aothr utot whh s dt wth thr or, th dd, ) = 3 Othrws dd, ) = Thrfor dd, ), so that th dsta btw ay two l vrts s atmost Cas-: Lt b utot vrts W osdr th followg subass Subas-: If th utots ar adjat, th th utot vrts otd by a l vrtx, h dd, ) = orrsodg to th l ar whh s odt wth both utots Subas-: If th utots ar oadjat If thr xsts a l whh s odt wth, th dd, ) = Othrws dd, ) = 3 both 0, IJMA All Rghts Rsrvd
B Basavaagoud*, Va R Dsa/ O th l-ut trasformato grahs / IJMA- 6), May-0 b l vrtx utot vrtx rstvly of If a l s odt wth Cas-3: Lt utot, th dd, ) = If a l s dt wth utot thr xst a l whh s odt wth utot oadjat to a l, th ar otd by a ath of lgth, h dd, ) = H from all th abov ass, dam ) s atmost 6 ACKNOWLEDEMENT * Ths rsarh s suortd by UC-MRP, Nw Dlh, Ida: F No -78/0 datd: 7-07-0 Ths rsarh s suortd by UC- Natoal Fllowsh NF) NwDlh No F/0-/NFO-0--OBC- KAR-873/SA-III/Wbst) Datd: Marh-0 REFERENCES M Aharya, R Ja, S Kasal, Charatrzato of l-ut grahs, rah Thory Nots of Nw York, 66 0) 3-6 B Basavaagoud, K Mrajkar, S Malgha, O lt grahs wth oarsss umbr o, Joural of Itllgt Systm Rsarh, 3 ) 009) - 3 B Basavaagoud, H P Patl, Jashr B Vragoudar, O th blok-trasformato grahs, grah quatos damtrs, Itratoal Joural of Advas S Thology, ) 0) 6-7 Chartr, H Hva, E B Jartt, M Shultz, Subgrah dsta grahs dfd by dg trasfrs, Dsrt Math, 70 997) 63-79 F Harary, rah thory, Addso-Wsly, Radg, Mass, 969) 6 V R Kull, M H Muddbhal, Lt grah ltat grah of a grah, J Aalyss Comut, ) 006) 33-3 7 V R Kull, Collg rah Thory, Vshwa Itratoal Publatos, ulbarga, Ida 0) 8 V R rsh, P Usha, Total domato lt grah, Itratoal J Math Comb, 0) 9-7 9 Va Rooj A C M, Wlf H S, Th trhag grah of a ft grah, Ata Mat Aad S Hugar, 6 96) 63-69 0 B Wu, J Mg, Bas rorts of total trasformato grahs, J Math Study, 3 00) 09-6 B Wu, X ao, Damtrs of jum grahs slf omlmtary jum grahs, rah Thory Nots of Nw York, 000) 3-3 Sour of suort: Nl, Coflt of trst: No Dlard [Coy rght 0 Ths s a O Ass artl dstrbutd udr th trms of th Itratoal Joural of Mathmatal Arhv IJMA), whh rmts urstrtd us, dstrbuto, rroduto ay mdum, rovdd th orgal work s rorly td] 0, IJMA All Rghts Rsrvd