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N mP g -NEG@@opW)o%pv@oQ p".)XiXiCY@YoYp\Oui~V@ oq@Oo#ep>ui0~ A~i89@ڂoFp,|@ !Eo{ +E@ Ro Riw\@\o \p|r}@oY'p\                              B 6h6h 0 uSh D5h 0Vhhhhhh A Si D-i 0K+j ASvjvj 0Dj D5 k BBk D5_k Bk D5k Bk D5l B8l D1Ul D5l B*l D5l B*m D5Dm B*ym D5m B*m D5n B77n B7nn Cn Cn Cn Cn C o C'o CAo C[o Cuo Co Co Cooooo D5o Bp D7/p D/fpPDFCreator0(9 Z6Times New Roman RegularX($USUS.,Y"6 b5:i+003|xULevel 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Math 203 eulerization handout(203 handout F06 6 ' $Leo Chouinard 0 .   *OLE**OLE*WPC..! .j  V  L    --v:--   --Z--   ----   --&--   -->&--   -->Z--   -->--   -->v:--   --~vB:--   --~B--   --~BZ--   --~&B--    -XXX--   -=XBXBX--   -`u``--   -B==--   - u  --   -u--   ---   -  Z Z--   -   --   ---   ---   -xxx--   -=xBxBx--   -```--   -B==--  -V%)  " )07=DKRY_fl rx~   --  -V%)")07=DKRY_flrx~ $).38=BHMRX]chntt}}--   -ZZ--  ----  Arial @ww 0w f- 12 rrThis vertex now odd (bad)         ----  "2 -XX-iThis vertex now    2 gX Xgeven (good)    ----  Arial @ww 0w f-  2 !A#   ----   2 rrB#   ----   2 rrC&   ----   2 cc D& "System f &-WPWin 6.0/OLE 1.0 Prefix Information Marker SnapGraphicsA0:@" ~$ ^PQRP@p`  @ 0 `Pp0 P@p` !""#@$0%%&`'P(()*p+,--.0/ 001P2@334p5`67789 :;;<@=0>>?`@PAABCpDEFFG0H IIJPK@LLMSP0pP0p P 0 p P0pP0pP0pP0 p!""P##$0%%p&''P(()0**p+,,P--.0//p011P223044p566P778099p:;;P<<=0>;FHP Color LaserJet 2600nwinspool- . /0ArialD$W|$W$W Arial9' v 'XX!! 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H1l2l0.G1j )7wWSW(SWSW#FF3456A#RWUWRWUW!!TWrxh" H1l2l0.G1j )7wWUW UWUW#FF3456B#SWWWSWWW""VWrh%# H1l2l01G1j )7wWWWVWxWW&FF3456C&UW*WUW##hXWbX  $ H1l2l0"0`1j )7wWxYWXW\YW FS"3456D& METAFILEPICT9p9u  V  L    --v:--   --Z--   ----   --&--   -->&--   -->Z--   -->--   -->v:--   --~vB:--   --~B--   --~BZ--   --~&B--    -XXX--   -=XBXBX--   -`u``--   -B==--   - u  --   -u--   ---   -  Z Z--   -   --   ---   ---   -xxx--   -=xBxBx--   -```--   -B==--  -V%)  " )07=DKRY_fl rx~   --  -V%)")07=DKRY_flrx~ $).38=BHMRX]chntt}}--   -ZZ--  ----  Arial @ww 0w f- 12 rrThis vertex now odd (bad)         ----  "2 -XX-iThis vertex now    2 gX Xgeven (good)    ----  Arial @ww 0w f-  2 !A#   ----   2 rrB#   ----   2 rrC&   ----   2 cc D& "System f &-WPC..! .$ T      --Da---   --D7---   --D---   --DS---   --D)---   --D---   --DuE---   --D---   --D---   --a---   --7---   -----   --S---   --)---   -----   --uE---   -----   -----   --ta---   --7t---   --t---   --tS---   --)t---   --t---   --utE---   --t---   --t---   --T$a---   --T7$---   --T$---   --T$S---   --T)$---   --T$---   --Tu$E---   --T$---   --T$---   --a---   --7---   -----   --S---   --)---   -----   --uE---   -----   -----   -,,,--  ---  ---  -<<<--  ---  -Cyyy--  -CC--  -C--  -Ckkk--  -C--  -C--  -C]]]--  -C--  -C--WPWin 6.0/OLE 1.0 Prefix Information Marker SnapGraphics\Q0a; ^PQRPIB;4-&yrk d ]  V O H A:3,%xqjc \UNG@92+~$wp !SPX`hp x(08@ H P X  ` hp x(08@HPX`hp x(08@H P!!"";FHP Color LaserJet 2600nwinspool- . /0Arialwww Arial9' v 'XX!! 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Gt((((w'((,-13,Ntarget.sgp $ -.F1l2l0" )s((P((:"3456,((,((w((]^Gq)-/20,Ntarget.sgp $ .0F1l2l0" )s(($((:"3456((t((w)+QQ*' DD)G(D((D(w*,+' DD)G((((w+-<<<,< DD)+GD((D((w,.MM-' DD )+)G(l((l(w-/G.'(G DD*,)*G($(($(w.0LL/( DD*,*Gl((l((w/1RR0( DD.$/&*G$(($(w021  DD ."/ G(h((h(w1L(213,Ntarget.sgp$ F1l2l0" )s(T((@(:"3456(`((0(w2 (?A(kU313,Ntarget.sgpF1l2l0" )s(((((:"3456 METAFILEPICTNDND$   ;&     --3---   --S&---   -----   --2---   --R'---   --v---   --v---   -----   --62 ---   --5 ---   --vKv---   --u1L---   --6R &---   --5 ---   --vRK&---   --uL---   -----   --qG---   -----   --Uk(---    ---   -1''--   -(//--   ---   -KK--   -```--   -L55--   - --   -2--   ---   ---   -  --   - --   -,,--   -T,----   -  --   -,S+S+--   -K<5<5<--   -LL--   -MM--   - Q  --   -`Q``--   - 1  --   -Q--   ---   -<<<--   -M--   -GG--   -L--   -R--   --- d(M3$ !USUS.,  XXXXVVVV)!dxdx(.3$ !USUS.,   Xba&OLE 2.0 Box <=8C HKKKK)!dxdx( $ Figure  1   ZXeYE<<C($$   1  gggg)!dxdx  diiii)!dxdxP Pdkkkk)!dxdx dmmmm)!dxdx dii)! dxdxpppp)!dxdx&0 d drrrr)!dxdx&P d dtttt)!dxdx&p d Pdvvvv)!dxdx&p Pd dxyxy)!dxdx3+0 d d d3+0 d d dd ddd(d2d*# tomakethatvertexeven,iftheotherendoftheedgewasanoddvertexalso,great,wevemadeboth .p+$ even.Butifitw`C3/b~0e  `.@EE   F wwaseven,wevenowmadeitodd,andneedtoduplicateanotheredgestartingatthatpoint H tofixit.Theprocesswillcontinueuntiltheedgeweduplicatehasanoddvertexatbothends.Seethe z2 diagrambelow,wherethedottededgewasjustaddedtofixvertexAatrightmiddle,butmessesupthe d vertexatupperright.Wemustnowaddanotheredgestartingatthere.Thatmeanswerebuildingapath      0  startingatvertexA,andthepathcanendonly  whenitgetstoanotheroddvertex.Whatthis B  saysisthat_eulerizing_Ԁrequirespairingupthe t , oddvertices,andconnectingthetwoodd  ^ verticesineachpairbyduplicatingapath   betweenthem.Clearlyeachsuchpathneeds    tobedoneinawaythatisasefficientas< $$ possible,butalsothepairingofoddverticesmustbedoneintelligently,tokeeptheaveragedistance n&  betweenverticesinthesamepairassmallaspossible.Intheabovediagram,theoddverticesoriginally X  areA,B,C,andD.WecanpairAwithBandCwithD,foranaveragepairdistanceof2,orwecanpair   AwithDandBwithC,andtheaveragepairdistanceisstill2!ButitwouldbesillytopairAwithCand  BwithD,foranaveragepairdistanceof3.Onesuresignevenbeforewefindtheaveragethatthelast 6 pairingisaloseristhattheedgefromBtoCcouldbeduplicatedtwicebythatpairing,onceduplicating h  anoptimalpathfromAtoCandagainduplicatinganoptimalpathfromBtoD.Ifyoueverfindone R edgeduplicatedtwiceinan_eulerization_,the_eulerization_Ԁisinefficient,becauseifbothduplicatesare  eliminated,theverticesallremainofevendegreeandthusthegraphisstill_eulerized_Ԁwithoutthem.  _c+f[KGj~71 << `X00000000@Ekb!PPd `;bd!0_ HIHIH$H$ HIHIHIHI HIHIHIHI HIHIHIHI HIHIHIHI HIHIHIHI HIHIHIHI H$H$HIHINotethatifyouaredealingwithagraphwhichisacompleterectangulararrayofedgeswithatleast2 %" edgesoneachside,thereisaparticularlysimplewaytofindabestpossible_eulerization_Ԁcalledtheedge $'# walkeralgorithm,describedinproblem#50onpages359360ofthetext. V(% Onceyouve_eulerized_Ԁthegraph,youcanfindanEulercircuitonthe_Eulerized_Ԁgraphandreinterpretthat *r'! asapathontheoriginalgraphthatefficientlymeetsyourneeds.Recallthatifyouvechosenthepairing +(" correctly,noedgeoftheoriginalgraphwillbeduplicatedmorethanonceinthe_eulerized_Ԁgraph,soan -)# edgeintheoriginalgraphwillbetraveledonatmostoneextratime. P.+$ Ї Example: Supposewewantareasonable_eulerization_Ԁofthefollowinggraph: H L@ *c`7/` 4 `.EH554c߈ 7 L9 Firstwemustidentifytheoddvertices.TheyaretheeightverticeslabeledAthroughHinthecopyofthe   graphbelow: 0  L@ *c`4/`  `.EHbF6 c߈ b  L!Notethatwellneed4pairs,andthateveryoddvertexexceptCandBisadjacenttoatleastoneotherodd  vertex(i.e.,connectedtoitbyasingleedge).AandDareadjacent,asareFandHandGandE.Infact, 3 EandHarealsoadjacenttoeachother,andthuseachisadjacenttotwooddvertices.Withfourpairs, e knowingeachpairwillrequireatleastoneedgetobeduplicated,wedliketokeepthenumberofedges O weduplicateascloseto4aspossible.YoumightthinkweshouldpairAwithD,EwithG,andFwithH   touseasmanyofthoseadjacentoddverticesaspossible,butifwedidthat,wedhavetopairBwithC,   andtheyresomewhatfarapart!itrequiresatleast5edgestogetbetweenthem,meaningwedneedto -  duplicate8edgesinall.Wecantrytodobetterthanthat,bycheckingwhetherwecanpairupBandC _  withothercloseroddverticeswithoutmessinguptoomanyotheroddvertices.Theclosestoddvertices  I  toBareA,D,andF,whichareeachtwoedgesdistant,andtheclosestoddverticestoCareEandG, !{ whicharealso2edgesdistant.IfwepairupBwitheitherAorD,wewillthenleavetheotherwithouta " nearbyoddvertexandhavemoreedgestoadd,butnotsoifwepairupBwithF.Likewise,ifwepairup '$  CwithE,weareleavingGwithoutanadjacentoddvertextopairwith,butifwepairCwithG,wecan Y%" stillpairEwithH.ThusourbestpairingisAwithD,BwithF,CwithG,andEwithH.Ourdistances &C# betweentheverticesinourpairsare1,2,2,and1,foratotalof6edgestoduplicate.Insomecases,there 'u$ canbeatieforthepairingthatproducesthebestpossible_eulerization_,butinthiscasewehavefoundthe (%  onlypairingrequiringjust6duplicatededges.One_eulerized_Ԁgraphforthispairinglookslike: !*& @ *e`,1!` L2 `.EHHH3 e߈ H  Notethatwithacomplexdiagramwithmanyoddvertices,itmaynotalwaysbeobviouswhatthebest y1  pairingis,butitwilloftenbeobviousthatcertainpairingsarenotgood.Ifthediagramiscomplex,we c  mayaskyoutofindadecentpairing,butwewontexpectyoutoproveyouvefoundthebestpossible   pairing.  Onefinalcomment:Whatchangesarenecessaryifwehaveaconnectedgraphwhereweneedtotravelall s+ oftheedgesbutwedontwantacircuit,justapath?I.e.,whatifwedontneedorwanttoendupwhere ] westart?Letsbespecific,supposewewishtostartatvertexAandendatadifferentvertexB,covering  everyedgeatleastonce.Theeasywaytofindanefficientrouteistostartbyintroducinganartificialedge    EbetweenAandBthatisspecialinthatedgeEcannotitselfbeduplicated.Now_eulerize_Ԁthegraphwith ;  edgeEadded,rememberingEcannotbeduplicated.Theresulthaseveryvertexofevendegree.Now m%  removeedgeEandvoila!,wehaveaconnectedgraphwhereAandBaretheonlytwoodddegree W  vertices.ThatmeansthereisnowanEulerpathgoingfromAtoBonthegraph,whichinterpretsasan   efficientpathtocovereveryedgeatleastonceontheoriginalgraph.  Problems: g  ! HHhH!1.Findgood_eulerizations_Ԁofeachofthefollowinggraphs,sothatyoucouldefficientlycovereachedge " andendupwhereyoustart. #  `RB>~"!  `.'@EH0% $Z^ ߀`TD@~"!7 p._'@E0%$F  /%!  .w+ `!^NJ~"! w `.##@EHI { wHI {         P   `%(L<8r~"! w `.'@ER I { ww *       X    (      `   })/"G73r^!CH  `4 4 '@E,IH 8 H H 6H  } H 2.0  In1partsb,c,ande,findanEulercircuitonthemodifiedgraphyoucreated.< $$ 3.FindagraphthatwouldbeusefulforcreatinganefficientpaththatstartsatvertexA,endsatvertexB X  andcoverseachedge(atleastonce)foreachofthefollowinggraphs.ThenfindanEulerpathstartingat   Aonthemodifiedgraph.  })13G73r^!  ` '@EH7  o Px }         P        X    ( )#*I95r^!/ p '@E(73 / /  4.0  Usingthe_eulerized_Ԁgraphs:#F $$   a)0  Inproblem1(e),indicateanefficientpathontheoriginalgraphthatstartsandendsintheupper $x! leftcornerandcoverseachedgeatleastonce,bynumberingtheedgesintheorderinwhichthey %" areused.$'#$$   b)0  Inproblem3(a),indicateanefficientpathontheoriginalgraphthatstartsatvertexAandendsat V(% vertexBandcoverseachedgeatleastonce,bynumberingtheedgesintheorderinwhichthey )@&  areused.