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Chapter10

EulerianandHamiltonianPaths

Circuits

Thischapterpresentstwowell?knownproblems?Eachofthemasksforasp ecialkind ofthetypesofpaths?EulerianandHamiltonian?havemanyapplicationsinanumb erofdi?eren Problem?TSP??anotherproblemwithgreatpracticalimp ortancewhichhastodowithcircuitswillb eexamined?10?1

Eulerpathsandcircuits

10?1?1TheKonisbergBridgeProblem

Eulergaveaformalsolutionfortheproblemand?asitisb elieved?establishedthegraphtheory?eldinmathematics? ?a?Konisbergbridge ?b?respectivegraph

Figure10?1:Konisb ergbridgeandthegraphinduced1

10?1?2De?ningEulerPaths

Obviously?theproblemisequivalentwiththatof?ndingapathinthegraphof?gure10?1?b?suchthatitcrosseseachedgeexactlyonce?Insteadofanexhaustivesearchofeverypath?Eulerfoundoutaverysimplecriterionforcheckingtheexistenceofsuchpathsinagraph?Asaresult?pathswiththisprop ertyhavehisname?De?nition

Ifthepathisclosed?wehaveanEulercircuit?Inordertopro ceedtoEuler?s

CheckingtheexistenceofanEulerpath

10?1?Euler?stheorem?

overticesofodddegree?Pr oof??

:AnEulercircuitexists?Astheresp ectivepathistraversed?eachtimewevisitavertexthereisavertexthroughanedgewecanleavethroughanotheredge?Forthestarting??nishingvertex?thisalsoholds?sincethereisoneedgeweinitiallyleavefromandanotheredge?throughwhichweformthecircle?Thus?wheneverwevisitano deweusetwoedges?whichmeansthatallverticeshaveevendegrees??

:Byinductiononthenumb erofvertices?Inductionb eginning:j

Vj?2?trivial?Inductionbasis:Supp oseforjVj?n

1?Selectanarbitraryvertexu

ttoit?Vertexuhadevendegreesoithasanevennumb erofneighb ors?No

walltheneighb orsofvhaveo dddegreesinceoneadjacentedgeisremoved?Bygroupingtheneighb orsincouplesandaddingoneedgeb etweeneachcouple?weobtainagraphwithnvertices?whereeveryvertexhasevendegree?Thus?thereexistsaneulercircuit?inductionbasis??When

anedge?u?w?addedb etweenneighb orsofvismetwhiletraversingthecircuit?wecanreplaceitbythepath?u?

v???v ? w??Thiswayeveryedgeinthegraphinitialistraversedexactlyonce?sothereexistsaneuleriancircuit?Incasewehavetwoverticeswitho dddegree?wecanaddanedgeb etweenthem?ob?tainingagraphwithnoo dd?degreevertices?Thisgraphhasaneulercircuit?Byremovingtheaddededgefromthecircuit?wehaveapaththatgo esthrougheveryedgeinthegraph?sincethecircuitwaseulerian?Thusthegraphhasaneulerpathandthetheoremisproved?2

?a?Graphwith eulercircuit ?b?Graphwitheulerpath ?c?Graphwithneithereulercir?

FindinganEulerPathThereareseveralwaysto?ndanEulerpathinagivengraph?Sinceitisarelativelysimpleproblemitcanb esolvedintuitivelyresp ectingafewguidelines:1?Alwaysleaveoneedgeavailabletogetbacktothestartingvertex?forcircuits?ortotheothero ddvertex?forpaths?asthelaststep?2?Don?t

arepresentedhere:Fleur y?salgorithm?G?V?E??

1cho osesomevertexu0ofG

2P?u03

considerP?u0 e 1 u1 e2???e i ui andcho oseanedgeei?1withthefollowingprop erties4 ?1?ei?1 joinsui withsomevertexui?1 and5 ?2?theremovalofei?1 do esnotdisconnectthegraphifp ossible6 addei?1 andui?1 inthepath7 removeei?18 ifP?W

9thenreturnP10else

11goto3Thealgorithmfor?ndinganEulerpathinsteadofacircuitisalmostidenticaltotheonejustdescrib ed?Theironlydi?erenceliesinstep1wherewemustcho oseoneofthetwoverticesofo dddegreeastheb eginningvertex?The?nalvertexofthepathwillb etheothero dd?degreevertex?3

Example:Figure10?3demonstratessomeimp ortantstepsinthepro cessdescrib edbythealgorithm?Sinceallverticeshaveo dddegreewearbitrarilystartfromtheupp erleftvertex?Thenumb ernexttoeachedgeindicatesitsorderintheEulercircuit?

?a?Thegraph ?b?Cannotavoidcross?ingabridge ?c?FullpathFigure10?3:Exampleof?eury?s algorithmexecutionEuleriangraphs?haveaveryimp ortantprop erty:Theyconsistofsmallerrings?Rings

arecycleswiththeadditionalrestrictionthatduringthetraversalofthecyclenovertexisvisitedtwice?Letusconsideraneuleriangraph?weknowthateveryvertexhasanevendegree?Anyringpassesthroughexactlytwoedgesadjacenttoanyofitsno des?Thismeansthatifweremovethering?theremainingofthegraphhasstillanevendegreeforallofitsno des?thusremainseulerian?Byrep eatingthispro cedureuntilnoedgesareleftwecanobtainadecomp ositionofaneuleriangraphintorings?Ofcourse?wecandecomp oseaneuleriancycletosmallercycles?notnecessarilyrings?butringshaveahigherpracticalvalue?intermsofnetworks?Thisisofhighimp ortanceinnetworkdesign?wherewewanttokeepanetworkaliveevenwhenanumb eroflinksaredown?Thisprop ertycanalsohelpasbuildtheeuleriancirclewiththeaidofthesmallrings?orcyclesagraphcanb edecomp osedto?Thepro cedurewefollowisdescrib edhereA

constructivealgorithmforbuildingeuleriancircuits?G?V?E??1cho osesomevertexu0 0 ?u0 u1 ???ui u0 andC1 ?v0 v1 ???vj v0

aremergedbytraversingoneofthemandinserttheotherwhenacommonvertexisfound?Theresultisanewcycle?ThecomputationalcomplexityofthisalgorithmisO?E??sinceweonlytraverseedgesuntilweformacycle?Forthemergingpro cedureO?E?timesu?cessinceonceagainwe4

a h gf de cb ?a?Thegraph a h gf de cb 1 5 432
?b?Firstcycle a h gf de cb 1 42
3 ?c?Secondcycle a h gf de cb 1 42
3 ?d?Thirdcycle a h gf de cb 1643
2 5 8 7 ?e?Amalgamatingsec?ondandthirdcycle a h gf de cb 143
2 58
7 6 11 10 9 13 12 ?f?Amalgamatingwiththe?rst

cycle?eulercy?cleFigure10?4:Exampleoftheconstructivealgorithmtraversethesetofedges?Wecaneasilymakethisalgorithm?ndeulerpaths?usingthesametrickasinEuler?stheorem?spro of?Theremustexistexactlytwoverticeswitho dddegree?otherwisenoEulerpathcanb efound?Weaddanedgeb etweenthesetwovertices?computeaneulercircuit?addobtainthepathbyremovingtheaddededge?Example:Figure10?4demonstratesthepro cessdescrib edbythealgorithm?Thereare3di?erentedge?disjointcyclesidenti?ed:a

?b?c?d?e?a?in?g?10?4?b???e?b?d?g

?e?in?g10?4?c??andf?e?h?g?f?in?g10?4?d???Wecanamalgamatethetwolatercyclestoobtainabiggercircle:f?e?b?d?g?e?h?g?f?in?g10?4?e???Thenthiscycleiscombinedwiththe?rstonegivingf?e?b?d?g?e?a?b?c?d?e?h?g?f?whichisanEulercycle??g10?4?f ???10?1?5

ExpansiontodirectedgraphsExpandingtodirectedgraphsisquitestraightforward?Asb efore?itisobviousthatifaneulercircuitexists?duringitstraversal?onemustalwaysvisitandleaveeveryvertex?Thismeansthatthenumb erofedgesleadingtoavertex?in?degree?mustb eequaltothenumb eroftheedgesthatleavethevertex?out?degree??Thistime?theconditionfortheexistenceofapathisslightlydi?erent?sinceforthe?rstvertexofourpathvwehavein

?deg ree?v??out ?deg r ee?v??1andforthelastvertexuin?deg r ee?u??out?deg r ee?u?? 1?Thatis5

b ecausewestartfromthe?rstvertexusinganout?goingedgeand?nishatthe?nalvertexthroughanin?comingedge?Sofordirectedgraphsthefollowingtheoremstands?Theorem10?2AdirectedgraphhasatleastoneEulercirclei?itisconnectedandforeveryvertexuin?de

??out?degree?s??1

?startingvertexofthepath?andin?degree?f??out?degree?f??1??nalvertexofthepath??Theeulercircuitsandpathscanb eobtainedusingthesamealgorithmsasb efore?onlythistimethedirectionofanedgeduringitstraversalmustb etakenintoconsideration?10?1?6

Applications

Euleriangraphsareusedratherextensively?asthey?renecessarytosolveimp ortantproblemsintelecommunication?parallelprogrammingdevelopmentandco ding?Moreo

ver?thecorresp ondingtheoryunderliesinmanyclassicmathematicalproblems?Inthenextsections?weexaminesomeinterestingexamplesLine

Drawings

Thisisamathematicalgame?wheregivenashap e?linedrawing?oneisaskedtore?pro duceitwithoutliftingthep encilorretracingaline?Y

Agraphhasaunicur saltr acingifitcanbetracedwithoutliftingthepencilorretracinganyline?Obviously?acl osedunicur saltr acingofalinedrawingisequivalenttoanEulercircuitinthecorresp ondinggraph?Similarly?anopen

unicur saltr acingequalstoanEulerpath?Thus?weendupwiththefollowingconditions:?

Alinedrawinghasaclosedunicursaltracingi?ithasnop ointsofintersectionofo dddegree?Alinedrawinghasanop enunicursaltracingi?ithasexactlytwop ointsofintersectionofo dddegree??In?gure10?5suchdrawingsapp ear?

?a?op en ?b?op en ?c?closedFigure10?5:Unicursal tracing6

10?1?7Eulerizationandsemi?Eulerization

IncaseswhereanEuleriancircuitorpathdo esnotexist?wemayb estillinterestedin?ndingacircuitorpaththatcrossesalledgeswithasfewretracededgesasp ossible?Eulerizationisasimplepro cessprovidingasolutionforthisproblem?Eulerizationisthepro cessofaddingduplicateedgestothegraphsothattheresultinggraphhasnotanyvertexofo dddegree?andthuscontainsanEulercircuit??Wecandothisbyselectingpairsofverticeswitho dddegreeandduplicatingtheedgesthatformapathb etweenthem?Foranyintermediatevertexweadd?duplicate?twoedgeskeepingitsdegreeevenifitwasevenando ddifitwaso dd?Atthisp ointwemustrecalltheprop ertyofanygraphthatthenumb erofverticeswitho dddegreeiseven?Thismeansthatnoo dd?degreevertexremainsuncoupled?Anexampleofanon?euleriangraphanditseulerizationapp earsin?gure10?6AsimilarproblemrisesforobtainingagraphthathasanEulerpath?Thepro cessinthiscaseiscalledSemi?Eulerizationandisthesameasb eforewiththeonlyadditionthatweaddedgesinsuchawaythattheinitialand?nalverticesofthepathhaveo dddegree?Thismeansthatifthevertexwewantthepathtostartfrom?orendto?hasevendegreewehavetoduplicatesomeedgessothedegreeb ecomeso dd?

?a?anon?euleriangraph

?b?EulerizationofthegraphFigure10?6:Eulerizationpro cessSomeworthmentionedp ointsare:1?Wecannotaddtrulynewedgesduringthepro cessofEulerizingagraph?Alladdededgesmustb eaduplicateofexistingedges?thatis?directlyconnectingtwoalreadyadjacentvertices??2?Duplicateedges?oftencalled?deadheadedges??canb econsideredasnewedgesorasmultipletracingsofthesameedge?dep endingontheproblemsemantics?3?Eulerizationcanb eachievedinmanywaysbyselectingadi?erentsetofedgestoduplicate?Wecandemandthattheselectedsetful?llssomeprop erties?givingbirthtomanyinterestingproblems?suchasaskingfortheminimumnumb erofedgestob eduplicated7

10?2HamiltonpathsandcircuitsAnotherimp ortantproblemhavingtodowithcircuitsandpathsisthesearchforacyclethatpassesthrougheveryvertexexactlyonce?Thismeansthatnotalledgesneedtob etraversed?Suchcycles?andtheresp ectivepaths?thatgothrougheveryvertexexactlyonce?arecalledHamiltoncircuits?pathandgraphsthatcontainhamiltoncircuits?arecharacterizedashamiltonian?De?nition10?4

Ahamiltoniancir cuitisacircuitthatstartingfromavertexu0 passesthroughal lotherverticesui exactlyonceandreturnstothestartingvertex?Ahamil tonianpathsimilarlyisapaththatstartingfromavertexu0

passesthroughal lotherverticesuiexactlyonceandstopsata?nalvertex?Theproblemof?ndingahamiltoncircuitorpath?isanNP?completeproblem?thusitishighlyunexp ectedto?ndap olynomialalgorithmforsolvingit?Thereexisthoweverseveralcriteriathatdeterminewhetheragraphishamiltonianornotforsomefamiliesofgraphs?Unfortunately?globalassumptionsuchashighdensity?oraguaranteedminimumdegreearenotenough?Wecaneasilyconstructanon?Hamiltoniangraphwhoseno des?minimumdegreeexceedsanygivenconstant?Whatifweuseavariableinsteadofaconstant?

ac1952? Everygraphwithn?3verticesandminimumdegreeatleastn?2hasaHamiltoncycle?Proof?LetG?V?E?b eagraphwithjN j?3and??G??n?2?Firstofall?thegraphisconnected?otherwisethedegreeofeveryvertexinthesmallercomp onentC wouldb elessthanjCj?n?2LetP?x 0 ???xk b ealongestpathinG?Thismeansthatallneighb orsofx0 andxk

lieonP?Otherwise?thepathcouldb eincreasedbyaddinganotalreadyincludedneighb or?whichcontradictsthemaximalityofP?Hence?atleastn?2oftheverticesx0

x1 ???xk? 1 areadjacenttoxk andatleastn?2ofthesameverticesxi forwhichxi?1 areneighb orsofx0 thatisadjacenttoxk andforwhichxi?1 isaneigh?b orofx0 ??gure10?7??ThenthecycleC?x0 ?xi?1 ?P?xi?1 :xk ??xi ?P?xi :x0

?formsahamiltoncycle?Thatisb ecausenoverticesexistthatarenotincludedinC?Iftherewasonesuchvertex?itwouldhavetob econnectedtoavertexinCsincethegraphisconnected?ThiswouldleadinalargerpaththanP?whichisacontradictiontoourhyp othesisthatPisalongestpath?Anothertheoremisbasedontheindep endencenumb era?G?ofagraphG?De?nition10?5

AnindependencesetV

0ofagraphG?V ? E?issubsetV

0jVforwhichholds?

Foranytwoverticesu? vofV

0?u? v?isnotanedgeinGDe?nition10?6

Theindependecenumbera?G?ofagraphG?V ? E?isthecardinalityofthelargestindependencesetofG8

XoXiXi+1

Xk PP1 2 3 4

?G??k?G?isthelargestintegerkforwhichGisk?connected?Now?withthede?nitionofindependencenumbergiven?wecanpro ceedandintro ducethetheorem?Theorem10?4Everygraphwithn?3verticesandk?G??a?g?hasaHamiltoncycle?Proof?Letk?G??kandCb ealongestcycleinG?WewillshowbycontradictionthatChastob eahamiltoncycle?soletCnotb eHamilton?First?weenumeratetheverticesinCcyclicallye?g?u

1 u2 ???ulquotesdbs_dbs17.pdfusesText_23