Independent Domination in Line Graphs

Transcription

1 Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju-1 1 ISSN Iddt Domato L Grahs M H Muddbhal ad D Basavarajaa Abstract - For ay grah G th l grah L G H s th trscto grah Thus th vrtcs of LG ar th dgs of G wth two vrtcs of LG adjact whvr th corrsodg dgs of G ar A domatg st D s calld ddt domatg st of LG f D s also ddt Th ddt domato umbr of LGdotd by L G quals m{ D; Ds a ddt domatg st of LG } A domatc artto of LG s a artto ofv L G all of whos classs ar domatg sts LG Th maxmum umbr of classs of a domatc artto of LG s calld th domatc umbr of LG ad dotd by d L G I ths ar may bouds o L G wr obtad trms of lmts of G but ot trms of lmts of LG Furthr w dvlo ts rlatosh wth othr dffrt domato aramtrs Also w troduc th coct of domatc umbr LG Subjct classfcato Numbr: AMS5-C69 C7 Ky words: Domatc umbr Domato umbr Grah Iddt domatg st Iddt domato umbr L grah ad Odd grah 1 INTRODUCTION: I ths ar w follow th otatos of [1] All th grahs cosdrd hr ar sml ft o-trval udrctd ad coctd As usual V ad q E dot th umbr of vrtcs ad dg of a grah G rsctvly I gral w us X to dot th sub grah ducd by th st of vrtcs X ad N v ad N v dot th o ad closd ghborhoods of a vrtx v Th otato 1 G G s th mmum umbr of vrtcs (dgs) a maxmal ddt st of vrtx (dg) of G Lt dg v s th dgr of vrtx v ad a vrtx of dgr o s calld a d vrtx ad ts ghbor s calld a suort vrtx MHMuddbhal Dartmt of Mathmatcs Gulbarga Uvrsty Gulbarga Karataa Ida E-mal: mhmuddbhal@yahooco D Basavarajaa Dartmt of Mathmatcs Gulbarga Uvrsty Gulbarga Karataa Ida ad Dartmt of Mathmatcs STJ Isttut of Tchology Rabur Karataa Ida E-mal: dbasavarajaa@gmalcom IJSER 1 htt://wwwjsrorg As usual G s th mmum (maxmum) dgr Th dgr of a dg uv of G s dfd by dg dg u dg v ad G G s th mmum (maxmum) dgr amog th dgs of G A sdr s a tr wth th rorty that th rmoval of all d aths of lgth two of T rsults a solatd vrtx calld th had of th sdr Lt b a tgr Th odd grah O s th grah whos vrtx st s V ad whch two vrtcs ar adjact f ad oly f thy ar dsjot as sts A l grah LG s th grah whos vrtcs corrsod to th dgs of G ad two vrtcs of LG ar adjact f ad oly f th corrsodg dgs G ar adjact A st D V s sad to b domatg st of G f vry vrtx ot D s adjact to a vrtx D Th domato umbr of G dotd by G s th mmum cardalty of a domatg st A st D s a ddt domatg st of G f D s also ddt Th ddt

2 Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju-1 ISSN domato umbr of G dotd by G s th mmum cardalty of a ddt domatg st Th coct of domato s ow wll studd grah thory (s [] [3]) Aalogously a st D V ( L( G )) s sad to b domatg st of LG f vry vrtx ot D s adjact to a vrtx D Th domato umbr of LG dotd by LG s th mmum cardalty of a domatg st LG A domatg st ddt domatg st f D of LG s sad to b D s also ddt Th ddt domato umbr of LG dotd by L G s th mmum cardalty of a ddt domatg st LG V L G A domatc artto LG s a artto of all of whos classs ar domatg sts LG Th maxmum umbr of classs of a domatc artto of LG s calld th domatc umbr of LG ad s dotd by d L G Th coct of domatc umbr G was troducd by Cocy tal [4] I ths ar may bouds o L G wr obtad Also thr rlatoshs wth othr domato aramtrs wr obtad Furthr w troduc th coct of domatc umbr of LG xact valus of domatc umbr wr obtad for som stadard grahs Also boud for domatc umbr s also obtad RESULTS: Itally w lst out ddt domato umbr of LG for som stadard grahs Thorm 1: a ( L( C)) / 3 ( L( K )) 1 b 1 ( L( K )) for m c m d ( L( K)) / ( ) ( L( W )) 3 Th followg Thorm rlats domato ad ddt domato LG Thorm : For ay coctd q grah ( L( ( L( q Equalty holds fg C4 D { v v v v } V ( L( G )) b Proof: Suos 1 3 th st of vrtcs whch covrs all th vrtcs LG Th D s a mmal - st of LG Furthr f th sub grah vrtcs 1 D cotas th st of v such thatdg v Th D tslf s a ddt domatg st of LG Othrws S D I whr D D ad I V ( L( D forms a mmal ddt domatg st of LG Sc V L G E G t follows that D S q Thrfor ( L( ( L( q For qualty suos G C4 th ths cas D S q / Clarly t follows that ( L( ( L( q I th followg Thorms w gv th ur bouds for ddt domato umbr of LG Thorm 3: If vry suort vrtx of tr s adjact to at last o d dg th ( L( T )) 1 whr m s th umbr of d dgs T Equalty holds for K star 1 1 F { } b th st of all d Proof: Lt 1 3 dgs T such that F gralty scv L T whr H m Now wthout loss of E T lt S F H F F ad H V ( L( F such that N[ F ] b th mmal st of vrtcs whch covrs all th vrtcs LT Clarly st of th vrtcs of a sub grah S s ddt th by th abov argumt S s a mmal ddt domatg st of LT Clarly t follows that 1 Thrfor ( L( T )) 1 S IJSER 1 htt://wwwjsrorg

3 Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju-1 3 ISSN Sc T Th ths cas K 1 Suos 1 L T K ad S 1 ( L( T )) 1 t follows that Thorm 4: For ay coctd grah G ( L( ( G ) 1 Proof: Suos 1 3 J { } b th maxmum st of dgs G such that N( ) N( ) for1 j Th j forms maxmal ddt st of dgs wth j ( G ) 1 ScV L G st D J H E G thr xsts a ddt whr J J ad H V( L( J such that H N[ J ] whch covrs all th vrtcs LG Clarly D forms a mmal ddt domatg st of LG ad t follows that D J Thrfor ( L( 1( G ) Thorm 5: For ay coctd q grah G ( L( q ( G ) C { v v v v } b th st of all Proof: Suos 1 3 o d vrtcs G Th thr xsts at last o vrtx v C whch s cdt wth at last o dg ( G ) G Now wthout loss of gralty LG suos 1 3 H { v v v v } b th st of all d vrtcs LG ad f V L G H I Th thr xsts a subst D I LG such that th sub grah D s ddt Clarly D s a - st of LG It follows that hc ( L( q ( G ) D q ad Corollary 1: For ay tr T cotag a dg ( T ) dam T 3 ( L( T ) q ( T ) f ad oly f Thorm 6: For ay coctd q grah G ( L( G )) 1 If ad oly f G cotas a dg E(G) such that dg q 1 j Proof: Assum that dg q 1 Now LG V L G E G suos F { v1 v v } b th st of all d vrtcs LG Th thr xsts a D { v v v v } V ( L( F vrtx st 1 3 whch covrs all th vrtcs LG Furthr f th sub grah D s ddt th D forms a mmal ddt domatg st of wth D a cotradcto LG Suos dg q 1 Th LG sc V L G E G thr xsts a vrtx v V ( L( whch covrs all vrtcs LG ad v D domatg st of LG Thrfor D 1 Clarly D tslf s a mmal ddt hc ( L( G )) 1 ad Thorm 7: For ay tr T ( L( ( 1) / Equalty holds f ad oly f T s somorhc to a sdr Proof: Lt st of all vrtcs wth F { v1 v v3 v} V ( L( b th F N( F ) whr F s th st of d vrtcs of LT Suos H N F Th H V ( L( F ad ( ) D I F whr I H LT Furthr f th sub grah ddt th D I F covrs all th vrtcs I F s forms a mmal ddt domatg st of LT Clarly t follows that I F ( 1) / Thrfor ( L( ( 1) / Suos T s somorhc to q sdr Th ths cas D F or D Sc for ay tr T q 1 ad ach T s odd t follows that ( L( ( 1) / I th followg Thorm w lst out th domatc umbr for som stadard grahs Thorm 8: I For ay comlt grah K IJSER 1 htt://wwwjsrorg

4 Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju-1 4 ISSN II d L K 1 f s v = f s odd For ay comlt bartt grah 1 vrtcs d L K = max 1 1 III For ay cycl C d L C 3 f s dvsbl by 3 IV For ay ath P d L P = othrws K 1 V For ay star K 1 wth 1 vrtcs d L K1 wth Proosto 1: Th domatc umbr of a l grah of O whr O s a odd grah s qual to 1 Proof: Th dg domatc umbr of a grah s vdtly qual to th domatc umbr of ts l grah [1] Th dgr of ach vrtx of l grah of O s most 1 Thrfor d L O 1 ad ths mls that ts domatc umbr s at Thorm 9: For ay coctd grahg d( L( 1 whr d L G s th domatc umbr of LG dgr of a vrtx of G ad (G) G s th mmum s th mmum dgr of a dg ofg Proof: Th umbr (G) s qual to th mmum dgr of a dg of th l-grah of G Accordg to [1] th domatc umbr of ths l grah caot b gratr tha (G) 1 Now w shall rov th frst qualty by th mthod of ducto If th d( L( dgr of ach vrtx of LG s gratr tha or qual to whr s a arbtrary ostv tgr th thr xsts a domatc artto of LG wth classs For 1 th rsult s tru Th rqurd artto cossts of o class qual to th whol V L G whch s vdtly a domatg st LG Now lt ad suos that th rsult s tru for 1 Cosdr a l grah LG whch th dgr of ach vrtx s at last Lt V L G b a maxmal (wth rsct to th st cluso) ddt st of vrtcs of LG Ths st s domatg; othrws a vrtx could b addd to t wthout volatg th ddc whch would b a cotradcto wth th maxmalty of LG V L G Lt b a l grah obtad from LG by dltg all vrtcs of V L G Each vrtx of LG s cdt at most wth o vrtx of V thrfor ach vrtx of LG has th dgr at last 1 Accordg to th hyothss thr xsts a vrtx domatc artto P of LG wth 1 classs Thrfor P V L G s a vrtx domatc artto of LG wth classs ad t mls that d( L( ( G ) For qualty w hav th followg Cass If G s a cycl Th d ( L( 1 c If G s a cycl Th d ( L( C wth s dvsbl by 3 C wth s ot dvsbl by 3 Fally w gv th followg Charactrzato Thorm 1: Lt T b a tr lt LT b th mmum dgr of a vrtx of LT ( ) Th d( L( ( L( T )) 1 Proof: Lt us hav th colors1 ( LT ( )) 1 ; w shall color th vrtcs of LT ( ) Frst w choos a trmal vrtx v of LT ( ) ad color t by 1 Now lt us hav a vrtx v of LT ( ) wth d vrtcs uw; suos that all vrtcs cdt wth w ar alrady colord Morovr f th umbr of ths vrtcs s lss tha ( L ( 1 w suos that thy ar colord by ar ws dffrt colors I th oost cas w suos that all colors 1 ( LT ( )) 1 occur amog th colors of ths IJSER 1 htt://wwwjsrorg

5 Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju-1 5 ISSN vrtcs Now w shall color th vrtcs adjact to u ad dstct from v W color thm th followg way If thr ar colors by whch o vrtx adjact to v s colord w us all of thm (Ths must b always ossbl accordg to th assumto) If th umbr of vrtcs to b colord s lss tha ( L ( 1 w color thm by ar ws dstct colors; th oost cas w color thm by usg all th colors1 ( LT ( )) 1 (t may form th color class) Th rsult s a colorg of vrtcs of LT ( ) by th colors 1 ( LT ( )) 1 wth th rorty that ach vrtx s adjact to vrtcs of all colors dffrt from ts ow If A for 1 ( L( T )) 1 s th st of all vrtcs of T colord by th th sts A1 A A form a domatc artto of LT ( ) LT 1 wth ( L( 1classs ad d( L( ( L( T )) 1 Accordg to Thorm 9 t d ( L( ( L( caot b gratr thrfor 1 REFERENCES: [1] F Harary Grah Thory Adso Wsly Radg Mass (1969) [] TWHays ST Hdtm ad P J Slatr Fudamtals of domato grahs Marcl Dr Ic Nw Yor (1998) [3] TWHays ST Hdtm ad P J Slatr Domato grahs advacd tocs Marcl Dr Ic Nw Yor (1999) [4] EJCocay ad ST Hdtm Towards a thory of domato grahs Ntwors 7 (1977) IJSER 1 htt://wwwjsrorg