Hamilton Paths and Hamilton Circuits Hamilton Path Hamilton
*Unlike Euler Paths and Circuits there is no trick to tell if a graph has a Hamilton Path or Circuit. A Complete Graph is a graph where every pair of vertices
A Search Procedure for Hamilton Paths and Circuits
direct the extension of the partial paths. KEY WORDS AND PHRASES: Hamilton path Hamilton circuit
Thomasons algorithm for finding a second hamiltonian circuit
The first hamiltonian path is obtained from the hamiltonian circuit containing yz by removing the other edge meeting y. Let q be the last vertex of P. One edge
On Hamiltonian Path and Circuits in Non-Abelian Finite Groups
09-Sept-2016 A Hamiltonian path that formed a cycle is called Hamiltonian circuit. (or Hamiltonian cycle) and the process of determining whether such paths ...
Hamilton Circuits/Graphs
Yes; this is a circuit that passes through each vertex exactly once. 3. Page 4. Hamilton Paths and. Circuits. Euler
The Mathematics of Touring (Chapter 6)
▷ By contrast an Euler path/circuit is a path/circuit that uses every edge Hamilton Paths and Hamilton Circuits. We can also make a Hamilton circuit ...
NP Completeness of Hamiltonian Circuits and Paths
24-Feb-2015 Now we will look at a proof that Hamiltonian circuits can be reduced to the vertex cover problem and then that Hamiltonian Paths can be reduced ...
8.3 Hamiltonian Paths and Circuits
Then G has a. Hamiltonian circuit if m ≥ ½(n2 – 3n + 6) where n is the number of vertices. Page 5. Hamilton Paths and Circuits. A. is a continuous path that
Antidirected Hamiltonian Paths in Tournaments* A simple path or
A simple path or circuit in a directed graph is said to be antidirected if every two adjacent edges of the path have opposing orientations in.
Hamiltonian Cycle 3-Color
https://courses.engr.illinois.edu/cs374/sp2021/scribbles/A-2021-04-27.pdf
Hamilton Paths and Hamilton Circuits Hamilton Path Hamilton
*Unlike Euler Paths and Circuits there is no trick to tell if a graph has a Hamilton Path or Circuit. A Complete Graph is a graph where every pair of vertices
8.3 Hamiltonian Paths and Circuits
Then G has a. Hamiltonian circuit if m ? ½(n2 – 3n + 6) where n is the number of vertices. Page 5. Hamilton Paths and Circuits. A. is a continuous path that
Polynomial Algorithms for Shortest Hamiltonian Path and Circuit
The problem of finding shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph belongs to the class of NP-Complete.
NP Completeness of Hamiltonian Circuits and Paths
Feb 24 2015 A graph G has a Hamiltonian Circuit if there exists a cycle that goes through every vertex in G. We want to show that there is a way to reduce ...
HAMILTON PATHS IN GRID GRAPHS* and 1 <-vy <=n}. A
contrast the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-complete. This provides a new
Lesson 4.5 - Hamiltonian Circuits and Paths
Hamiltonian path. If the path ends at the starting vertex it is called a. Hamiltonian circuit. Try to find a Hamiltonian circuit for.
Thomasons algorithm for finding a second hamiltonian circuit
circuit (i.e. when there is an edge in the graph from the end of the hamiltonian path to y). On cubic graphs Thomason's algorithm is finite and completely
Graph Theory Eulerian and Hamiltonian Graphs
Hamiltonian Circuit: A Hamiltonian circuit in a graph is a closed path that visits every vertex in the graph exactly once. (Such a closed loop must be a cycle.).
A Search Procedure for Hamilton Paths and Circuits
A search procedure is given which will determine whether Hamilton paths or circuits Hamilton path Hamilton circuit
Chapter 6: Graph Theory
Circuit: a path that starts and ends at the same vertex looking at a graph if it has a Hamilton circuit or path like you can with an Euler.
[PDF] Hamilton-Paths-and-Hamilton-Circuits-1pdf
A Hamilton Path is a path that goes through every Vertex of a graph exactly once A Hamilton Circuit is a Hamilton Path that begins and ends at the same vertex
[PDF] 83 Hamiltonian Paths and Circuits
A Hamilton circuit (path) is a simple circuit (path) that contains all vertices and passes through each vertex of the graph exactly once • How can we tell if a
[PDF] Eulerian and Hamiltonian Paths Circuits - Csduocgr
Eulerian and Hamiltonian Paths Circuits This chapter presents two well-known problems Each of them asks for a special kind of path in a graph
[PDF] Eulerian and Hamiltonian Paths - Csduocgr
Definition 1: An Euler path is a path that crosses each edge of the graph exactly once If the path is closed we have an Euler circuit In order to proceed to
[PDF] Hamilton Circuits/Graphs
Definition: A simple path in a graph G that passes through every vertex exactly once is called a Hamilton path and a simple circuit in a graph G that passes
[PDF] Lesson 45 - Hamiltonian Circuits and Paths
In honor of Hamilton and his game a path that uses each vertex of a graph exactly once is known as a Hamiltonian path If the path ends at the starting vertex
[PDF] Hamilton Paths and Hamilton Cycles
A Hamilton cycle in a graph G is a closed path that passes through each vertex exactly once and in which all the edges are distinct Definition A Hamiltonian
[PDF] Hamilton circuits (Section 22)
We prove there is a Hamilton circuit by induction Let pm be the statement “As long as m + 1 ? n there is a path visiting m + 1 distinct vertices with no
[PDF] Euler Paths Planar Graphs and Hamiltonian Paths
A set of nodes where there is an path between any two nodes in the Very hard to determine if a graph has a Hamiltonian path
[PDF] hamiltonian path
hamiltonian path • A Hamiltonian path of a graph is a path that visits every node of the graph exactly once • Suppose graph G has n nodes: 12 n
[PDF] Hamilton-Paths-and-Hamilton-Circuits-1pdf
A Hamilton Path is a path that goes through every Vertex of a graph exactly once A Hamilton Circuit is a Hamilton Path that begins and ends at the same vertex
[PDF] 83 Hamiltonian Paths and Circuits
A Hamilton circuit (path) is a simple circuit (path) that contains all vertices and passes through each vertex of the graph exactly once • How can we tell if a
[PDF] Eulerian and Hamiltonian Paths Circuits - Csduocgr
Eulerian and Hamiltonian Paths Circuits This chapter presents two well-known problems Each of them asks for a special kind of path in a graph
[PDF] Eulerian and Hamiltonian Paths - Csduocgr
Definition 1: An Euler path is a path that crosses each edge of the graph exactly once If the path is closed we have an Euler circuit In order to proceed to
[PDF] Hamilton Circuits/Graphs
Definition: A simple path in a graph G that passes through every vertex exactly once is called a Hamilton path and a simple circuit in a graph G that passes
[PDF] Lesson 45 - Hamiltonian Circuits and Paths
Hamiltonian path If the path ends at the starting vertex it is called a Hamiltonian circuit Try to find a Hamiltonian circuit for
[PDF] Hamilton Paths and Hamilton Cycles
Hamiltonian Graphs Definition A Hamilton path in a graph G is a path that contains each vertex of G exactly once Definition A Hamilton cycle in a graph
[PDF] Hamilton circuits (Section 22)
vn ? v1 is a Hamilton circuit since all edges are present “As long as m + 1 ? n there is a path visiting m + 1 distinct vertices with no
[PDF] Euler Paths Planar Graphs and Hamiltonian Paths
Degree of node A ? The number of edges that include A ? Strongly Connected Component ? A set of nodes where there is an path between any two nodes in
[PDF] hamiltonian path
monotone circuit value is circuit value applied to monotone circuits c 2011 Prof Yuh-Dauh Lyuu National Taiwan University Page 261 Page 44
Chapter10
EulerianandHamiltonianPaths
Circuits
Thischapterpresentstwowell?knownproblems?Eachofthemasksforasp ecialkind ofthetypesofpaths?EulerianandHamiltonian?havemanyapplicationsinanumb erofdi?eren Problem?TSP??anotherproblemwithgreatpracticalimp ortancewhichhastodowithcircuitswillb eexamined?10?1Eulerpathsandcircuits
10?1?1TheKonisbergBridgeProblem
Eulergaveaformalsolutionfortheproblemand?asitisb elieved?establishedthegraphtheory?eldinmathematics? ?a?Konisbergbridge ?b?respectivegraphFigure10?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?sCheckingtheexistenceofanEulerpath
10?1?Euler?stheorem?
overticesofodddegree?Pr oof??:AnEulercircuitexists?Astheresp ectivepathistraversed?eachtimewevisitavertexthereisavertexthroughanedgewecanleavethroughanotheredge?Forthestarting??nishingvertex?thisalsoholds?sincethereisoneedgeweinitiallyleavefromandanotheredge?throughwhichweformthecircle?Thus?wheneverwevisitano deweusetwoedges?whichmeansthatallverticeshaveevendegrees??
:Byinductiononthenumb erofvertices?Inductionb eginning:jVj?2?trivial?Inductionbasis:Supp oseforjVj?n
1?Selectanarbitraryvertexu
ttoit?Vertexuhadevendegreesoithasanevennumb erofneighb ors?Nowalltheneighb 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?W9thenreturnP10else
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?Ringsarecycleswiththeadditionalrestrictionthatduringthetraversalofthecyclenovertexisvisitedtwice?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 v0aremergedbytraversingoneofthemandinserttheotherwhenacommonvertexisfound?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?Thatis5b 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 tracing610?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 tonianpathsimilarlyisapaththatstartingfromavertexu0passesthroughal 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 andxklieonP?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?isthecardinalityofthelargestindependencesetofG8XoXiXi+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
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