Drete rdo wl wth rrer o loll fte grh Theo Ue Aterd Shool of Teholog Weeerde 9 97 DZ Aterd The etherld El: te@hl Atrt We ot eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl o loll fte grh the reee of ltle fto rrer O eh edge of the grh d eh rrer there re ef rolte defed Ele of ltle fto rrer grh d lto o the teger re ge Keword: Rdo wl orto refleto rrer grh thet Set Clfto: Prr 6G5 Seodr 6J5 Itrodto Rdo wl e ed ro dle: eoo to odel hre re d ther derte ede d olog where org rrer ge trl odel for wde ret of heoe h lfed odel of Brow oto eolog to dere ddl l oeet d olto d ttt to le eetl tet roedre oter ee to ette the e of the World Wde We g rdoed lgorth Bro d C 5 ge reew of rdo wl o grh where the geerlto of the oet of deo to hoogeeo trtre g fte grh odered Drh Joo d Wheter 6 deelo tehe to ot rgoro od o the ehor of rdo wl o o Ug thee od the llte the etrl deo of rdo o wth fte teeth t rdo oto or teeth wth rdo t fte legth Rdo wl he ee tded for dede o reglr trtre h ltte We ow ge ref htorl reew of the e of rrer oe-deol drete rdo wl Weel 96 ded the ll role of rdo wl retrted etwee refletg d org rrer Ug geertg fto he ot elt ereo for the rolt of orto Leher 963 tde oe-deol rdo wl wth rtll refletg rrer g otorl ethod Gt 966 trode the oet of ltle fto rrer f: tte tht or reflet let throgh or hold for oet D Khdlr d Se 976 fd the rte geertg fto of the rolte of rtle rehg ert tte der dfferet odto Per 985 oder etr rdo wl wth oe or two odre o oe-deol ltte At the odre the wler ether ored or refleted to the te Ug geertg fto the rolt dtrto of eg t oto fter te oted well the e er of te efore orto El-Shehwe ot orto rolte t the odre for rdo wl etwee oe or two rtll org odre well the odtol e for the er of te efore tog ge the orto t efed rrer g odtol rolte I th er we ot eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl wth ltle fto rrer edg the theor two dreto: A the tte e o loger oe-deol ltte or do loll fte grh B oe or two org or refletg rrer re reled fte et of ltle fto rrer
Theo Ue We hooe trtre of or grh tht hdle wth the well ow ltte trtre well trtre le tr grh fte d fte d le grh Or grh ot of ltle fto rrer erte tte o the edge etwee the f d fte et of hlf le eh wth fte er of tte trtg eh rrer Eh hlf le h toologl ed O eh edge of the grh rdo wl wth t ow terl tte d g rolte troded Whe the wler rehe ltle fto rrer rdo wl tted ordg to etr et of rolte or the rtle ored the rrer Eh rrer d eh edge h t ow rolt reter I eto we e geertg fto to fd the eeted er of rrl to tte the rolt of orto d the eeted te efore orto I eto 3 we le oe ele of grh wth ltle fto rrer: two tr grh d le grh I the lt eto we l or theor l tto: the et of teger I ede A d B we ge roe of oe relt eto A grh wth ltle fto rrer Derto of the rdo wl wth ltle fto rrer I grh we he erte rereetg the f Betwee d there rdo wl wth fte er of tte whh we er 3 the dreto fro to whe We wll e the reto for the edge etwee d Eh rdo wl fro to h t ow reter d rr where the oe-te forwrd rolt oe-te wrd d r-- the rolt to t for oet the e oto We ded for eh d It lo ole to oe fro log hlf le wth tte 3 A hlf le trtg lelled d h ed Ω I there rolt to oe oe te the dreto of rolt to t for oet rolt for edte orto d rolt to oe oe te the dreto Ω where ee lo fg Ω tte: 3 oe te rolte: r tte: r Fgre Derto of the rdo wl
Drete rdo wl wth rrer 3 We trt ether d o edge d d where e: trt d trt d rereeted d or we trt o edge h where e: trt Eeted er of rrl We re tereted the eeted er of rrl the f well the eeted er of rrl the other tte of or grh We defe: trt fter te tte Pte f ot f for f trt rrl eeted er of rrl trt eeted er of O terl : d O hlf le : We trt o the terl d or o the hlf le h Theore If the: the e olto of Q where f f Strtg o terl d: d d Q δ δ δ Strtg o hlf le h: h f h f Q δ δ O terl we he: If d the:
4 Theo Ue the : If d O hlf le we he: the: h If h h h h h the : h If f f If the we ot lr relt lg l Hotl rle Proof See Aed A 3 Prolt of orto We re tereted the rolt of orto f well the ed of the hlf le We defe: trt fter te f Pte The rolt of orto f ge Aother w to detere the rolt of orto f ge the et orollr of Theore Alto of th orollr e fod eto 3 d 4
Drete rdo wl wth rrer 5 Corollr Porto f t or wth hlf le trt o f we erl o o hlf le wth we trt f Proof Ug Theore we get: Q f we trt o hlf le wth otherwe Q The e theore ge: o: Q Le Ge r rdo wl o the o-egte teger wth rtl refletg rrer trt d ed Ω we he: rrl eeted er of orto Ω P Proof I we te refletg rolt d orto rolt Solg the relet dfferee eto we fd: Ug th relt we lo get orto P Ω We re ow red for: Corollr If o hlf le wth ed Ω we he the: Ω o Port h f h f h Proof Ue Theore d Le where tte Le ow odered the oe ot otrto of or olete grh or hlf le We wll e th orollr eto 3 d 4 Rer Corollr oeee of orollr
6 Theo Ue 4 Eeted te efore orto We re tereted the eeted te efore orto We defe: eeted te efore orto whe trtg o terl eeted te efore orto whe trtg o hlf le eeted te efore orto whe trtg Theore If o eh hlf le wth we he the: the e olto of Λ where: d f d f d f Λ O terl : O hlf le : Proof See ed B
Drete rdo wl wth rrer 7 3 Ele of ltle fto rrer grh 3 A fte tr grh We oder tr grh wth the etre d ll edge re of the terl te we trt o edge tte rolte r Fgre A fte tr grh I fte tr grh we he: f { } d otto: r r We he: 3 Eeted er of rrl To ot the eeted er of rrl or fte tr grh we e Theore : Q Ited of the eto wth we wll e: Th eto el dered fro Q orto or fte grh If we fd: ς ς ee lo orollr or oerg tht we del wth the rolt of 3 wth ς where ς 3 9
8 Theo Ue If : wth 3 ς ς ς ς I oth e we he : o ς ς ς ζ If we trt the eter of the tr grh we he: ς ς 3 e orto te the reee of org tte the edot We ow oder tr grh wth the etre d org tte the edot : } { f We trt otto: r r We defe for : R W Theore ge ow: Λ R W
Drete rdo wl wth rrer 9 33 A el e We get el fte tr grh whe we te d tte rolte r tte rolte r Fgre 3 A el e Ug eto 3 we get the eeted er of rrl: ς ς Iterretto: rdo wl o -AB wth dfferet rolte left d rght fro the trtg ot d three f : -A d B A B Ug eto 3 we ot the e orto te trtg the reee of org tte the edot: Iterretto: rdo wl o -AB wth dfferet rolte left d rght fro the trtg ot f d two org rrer: -A d B A B W R Rer If we get the well ow relt: AB ee Feller 349 3 A fte tr grh We oder tr grh wth gle f the etre d ll edge re of the hlf le te we trt o the edge otto : r r Ω tte rolte r Ω Ω Fgre 4 A fte tr grh
Theo Ue 3 Eeted er of rrl d rolt of rrl To ot the eeted er rrl d rolt of rrl we l Theore : f f where whh led to: f f Ag g Theore we fd: O hlf le : O hlf le 3 : f f We defe f rolt to t whe trtg A well ow relt : f If o hlfle d : f
Drete rdo wl wth rrer 3 Aorto rolte Bede orto there the olt of orto Ω whe Whe ge: o: Porto ow we oe tht there t let oe wth: Ug 9 wth org tte d tg we fd for f the ζ d f f We frt oder the e : Th led to: f the: Porto Porto We he: Porto et we le the e : Ω Porto Porto Th d g g 9 wth org tte led to: d f : Ω 3 Porto Ω Porto Ω Here we lo he: Porto Porto 3 Ω All relt th eto lo e erfed lg orollr the f e d orollr the Ω tto
Theo Ue 33 e orto te Alg Theore we ot the e orto te f whe trtg If : The to e derget For the e orto te tte o hlf le we fd: 34 Sel e: two hlf le trtg f A oeee we he ow lo led rdo wl o wth gle f d two reter d rght d left fro the org: te two edge reer the tte of the eod edge -- d e reter o the frt d o the eod edge 33 A Pote Oreted Cle Grh We he rrer le grh: We trt d defe rtfl rrer whh the e rrer A ote oreted le grh defed : otto : tte rolte r Fgre 5 A le grh 33 Eeted er of rrl d rolt of rrl Theore ge ow:
Drete rdo wl wth rrer 3 Pt for : where We ow he: where I fte grh we he: o: the o f he : we For We lo he: f 33 e orto te Ug Theore we ot: Λ Λ Λ µ µ µ o : µ
4 Theo Ue 4 Alto of ltle fto rrer grh 4 Itrodto Theore d lo e ed lg rdo wl wth ltle fto rrer o et of the teger Eg rdo wl o hlf le wth rtll refletg rrer e led g le etwor wth oe f d oe ed Ω I the et eto we wor ot other lto of or theor 4: Rdo wl o the teger wth two f We oder drete rdo wl o the teger of the --r te r wth f d I f we he rolte r r d f we he rolte r r We dere th rdo wl grh wth three ooet: the frt oe fte terl where the tte re ered fro to : wth The eod ooet hlf le trtg tte re ered o th ooet th orreod wth or orgl tte e The lt ooet hlf le trtg tte re ered o th hlf le whh orreod wth - the teger O the terl d the frt hlf le we he reter o the lt hlf le we e reter to get the dered rdo wl Ω r r Ω Fgre 6 Rdo wl o the teger wth two f Frt we hdle wth We e Theore whe trtg Solg the two eto we fd: Ug orollre d : Port o Porto Porto Ω Porto Ω f : Porto f f : Porto f
Drete rdo wl wth rrer 5 The eod rt of Theore ge: f f f f the lt for le re dted o ehlf of dfferet erg of the tte e d reter o the egte teger et we e Theore whe trtg : orgl tte e We td the e roeed log the e le d fd: ow we ot the rolt of rrl f If d the: f Th the forl grh lgge for the orgl tte e we eed to hge - ow we roeed wth the e We e Theore whe trtg olg the two eto we fd:
6 Theo Ue Porto Porto et we e Theore whe trtg If d the: orgl tte e We get: f Th the forl grh lgge for the orgl tte e we he to hge - Bee of the ft tht ot oth d e le th we he: Ζ 5 Colo Ug geertg fto we oted eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl o loll fte grh the reee of ltle fto rrer eto Elt olto were oted for oreted le grh fte d fte tr grh eto 3 We lo got relt the feld of oedeol rdo wl: o the teger wth two rrer eto 4 o terl wth three rrer d dfferet rolte etwee the rrer eto 33 d o the teger wth oe rrer d dfferet rolte left d rght fro the rrer eto 34
Drete rdo wl wth rrer 7 Aed A A Proof of theore Ce : or The rdo wl etwee d d d d the rdo wl o hlf le h e dered the dfferee eto: r wth hrtert eto: r Bee of or we he: wth The rdo wl etwee d d d the rdo wl o h e dered the dfferee eto: r δ wth olto : 4 4 r r 3 We frt loo t the terl e B fog o tte d etwee d d d we get: r r 4 Ug d 4 we ere : d d 5 Proeedg log the e le t ow etwee d d ge: 4 r 6 4 r 7 We ow oder hlf le wth h: Ug wth d r we get: 8 After oe llto we ot o hlfle h:
8 Theo Ue 4 4 r r 9 We ow fo o d t eghor: Frt we hdle the terl Whe d the e : Stttg the forle we fod for d 5 we get: d: Both forle re ld for t we eed the lt oe wth terhgg d ge: Whe g 36 d 7: 3 Whe d: 4 For the hlf le wth g 8: 5 Hlf le wth g 9: 6 We re ow red for the fl rt For d we he g d 5:
Drete rdo wl wth rrer 9 I the terl e we get ddtol ter whe e 3 d whe d e 4 Whe we get the hlf le e ddtol ter e 6 We get the relt of theore tg d otg tht: the If Ce : d Ug the e ethod e t ow wth d ted of d we fd o terl : the : If d the : If d where O hlf le we he: the : If d wth the : If d where For we he: We get the e wer g d l Hotl rle e Whe trtg or d o the terl d we get the e relt wth ddtol ter whe d whe d Strtg o h we get ddtol ter
Theo Ue Aed B B Proof of Theore The rdo wl etwee d d d the rdo wl o h e dered the dfferee eto: r 7 Frt we d the terl rt: Iterl e A olto of 7 ge : 8 Ug 8 wth d we ere d d Ug tht ereo we get: We e the lt forl whe o we he to terhge d the forl: Iterl e Followg the e ethod e we get: The e forle re fod lg l Hotl rle twe the terl e et we d hlf le For hlf le wth we get: Fll we e:
Drete rdo wl wth rrer Referee Feller W 968 A Itrodto to rolt theor d t lto thrd edto Vol Joh Wle ew or Weel B 96 The rdo wl etwee refletg d org rrer A th Sttt 3 765-769 3 Leher G 963 Oe-deol rdo wl wth rtll refletg rrer A th Stt 34 45-4 4 Gt H C 966 Rdo wl the reee of ltle fto rrer Jor th S 8-9 5 D S Khdlr S d Se K 976 A odfed rdo wl the reee of rtll refletg rrer J Al Pro 3 69-75 6 Per O E 985 Phe trto oe-deol rdo wl wth rtll refletg odre Ad Al Pro 7 594-66 7 El-Shehwe A Aorto rolte for rdo wl etwee two rtll org odre I J Ph A: th Ge 33 95-93 8 Bro R d C D 5 Rdo wl o grh: de tehe d relt J Ph A: th Ge 38 R45-R78 9 Drh B Joo T d Wheter J 6 Rdo wl o o J Ph A: th Ge 39 9-38