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Year 12Mathematics

Contents

uLake Ltdu a e tduLake LtdInnovative Publisher of Mathematics Texts

• AchievementStandard .................................................. 2

• GraphTheoryBasics .................................................... 3

• EulerPathsandCircuits................................................. 6

• EvenandOddVertices................................................... 10

• Traversability.......................................................... 13

• HamiltonianCircuitsandPaths........................................... 18

• TravellingSalesmanProblem............................................. 22

• ShortestPathsandDijkstra’sAlgorithm.................................... 27

• MinimumSpanningTrees ............................................... 37

• Kruskal’sAlgorithm .................................................... 38

• TheChinesePostmanProblem............................................ 47

• TheKönigsbergBridgeProblem .......................................... 49

• PracticeInternalAssessment ............................................. 52

• Answers............................................................... 56

Robert Lakeland & Carl NugentUse Networks in Solving ProblemsIAS 2.5

IAS 2.5 – Year 12 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent

2 IAS 2.5 – Networks

NCEA 2 Internal Achievement Standard 2.5 – NetworksThisachievementstandardinvolvesapplyingnetworkmethodsinsolvingproblems.

◆ ThisachievementstandardisderivedfromLevel7ofTheNewZealandCurriculum, andisrelatedtotheachievementobjective: ❖ chooseappropriatenetworkstofindoptimalsolutions intheMathematicsstrandoftheMathematicsandStatisticsLearningArea.

◆ Applynetworkmethodsinsolvingproblemsinvolves: ❖ selectingandusingmethods ❖ demonstratingknowledgeofconceptsandterms ❖ communicatingusingappropriaterepresentations.

Relationalthinkinginvolvesoneormoreof: ❖ selectingandcarryingoutalogicalsequenceofsteps ❖ connectingdifferentconceptsorrepresentations ❖ demonstratingunderstandingofconcepts ❖ formingandusingamodel; andalsorelatingfindingstoacontext,orcommunicatingthinkingusingappropriatemathematical statements.

Extendedabstractthinkinginvolvesoneormoreof: ❖ devisingastrategytoinvestigateasituation ❖ identifyingrelevantconceptsincontext ❖ developingachainoflogicalreasoning,orproof ❖ formingageneralisation; andalsousingcorrectmathematicalstatements,orcommunicatingmathematicalinsight.

◆ Problemsaresituationsthatprovideopportunitiestoapplyknowledgeorunderstandingof mathematicalconceptsandmethods.Situationswillbesetinreal-lifeormathematicalcontexts.

◆ Methodsincludeaselectionfromthoserelatedto: ❖ shortestpath ❖ traversability ❖ minimumspanningtree.

Achievement Achievement with Merit Achievement with Excellence• Applynetworkmethodsin

solvingproblems.• Applynetworkmethods,

usingrelationalthinking,insolvingproblems.

• Applynetworkmethods,usingextendedabstractthinking,insolvingproblems.

A B C

F E D

G H I

Start

End

3IAS 2.5 – Networks


Introduction

Graph Theory BasicsThegraphswewillstudyinthisAchievementStandardaredifferentfromthoseyouhavepreviouslystudiedanddrawnusinggridpaper.InGraphTheoryagraphconsistsofasetofvertices(nodes)andedges(straightorcurvedlines).Eachedgejoinsonevertex(node)toanother.

Thedegreeofavertex(node)isthenumberofedges(lines)whichstartorendatthevertex.

VertexAhasdegree3becauseithasthreeedges(lines)‘comingorgoing’fromitwhereasvertexChasdegree5becauseithasfiveedges(lines)‘comingorgoing’fromit.Insomegraphsitispossiblethattwolineswillcrossbutunlesstheyaremarkedasavertextheyshouldnotbetreatedassuch.Seethediagrambelow.TheintersectionoflinesACandDBisnotavertex.

Vertices(nodes)

Edges(lines)

Graph

A

B

DC

A

BD

C

Degree3

Degree5

Think of two lines that cross, but not at a vertex, as similar to an underpass on a motorway.

In this Achievement standard reference is made to graph theory and graphs. Networks are an application of graph theory. The authors use the term, graph, as this is the accepted practice, but it is possible to call these graphs, networks, in order to differentiate them from Cartesian graphs.©

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Hamiltonian Paths and Circuits

AHamiltonianpath,namedaftertheIrishmathematicianSirWilliamRowanHamilton,isonethatvisitseachvertex(node)onlyonce.Eachedgedoesnothavetobetraversed.AHamiltoniancircuitorcycleisonethatvisitseachvertex(node)onlyoncebutalsoreturnstoitsstartingvertex(node).EachedgedoesnothavetobetraversedinaHamiltoniancircuit.AnexampleofaHamiltonianpathisdrawnbelow.

Apossiblepaththatvisitseachvertex(node)onlyoncebutdoesnotreturntothestartisdrawnbelow.NotealsothatnotalledgeshavebeentraversedwhichisnotarequirementofaHamiltonianpath.

AnexampleofaHamiltoniancircuitisdrawnbelow.

Apossiblepaththatstartsandendsatthesamepointandvisitseachvertex(node)isdrawnontheright.NotealsothatnotalledgeshavebeentraversedwhichisnotarequirementforaHamiltoniancircuit.

Hamiltonian circuits are useful in applications where it is necessary to visit each ‘vertex’ for example, a salesman’s clients in a town.

A circuit (cycle) is a closed path, i.e. it finishes where it starts.

A

BD

C

A B C

F E D

G H I

A

BD

C

Start/end

1 2

34

Unfortunately there is not an equivalent formula to Euler’s to identify whether or not a graph has a Hamiltonian path or circuit.

In an Euler path you can visit each vertex (node) more than once but each edge only once. In a Hamiltonian path you can only visit each node once.

A B C

F E D

G H I

Start

End

In problems involving Hamiltonian circuits there are many equivalent reversal routes. When we don’t want to count reversal routes we use the term ‘distinct’ routes.

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37.

DrawtheHamiltonianpathorcircuitbelow.Labelyourstartandendpoints. Alinethatyoumarkassolidwillbeonethatyouneednottraverse.

A

B

C

D

E

F G

A

B

C

D

E

F G

38.A

B

C

D

E

F

G

H

DrawtheHamiltonianpathorcircuitbelow.Labelyourstartandendpoints. Alinethatyoumarkassolidwillbeonethatyouneednottraverse.

A

B

C

D

E

F

G

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39.

A B C D

EFGH DrawtheHamiltonianpathorcircuitbelow.Labelyourstartandendpoints. Alinethatyoumarkassolidwillbeonethatyouneednottraverse.

A B C D

EFGH

Circuitorpath?

Justification:

Circuitorpath?

Justification:

Circuitorpath?

Justification:

Path/Circuit:

Path/Circuit:

Path/Circuit:

I

I




Everysuburb(node)ofourgraphisnowtouchedbyalink(fibreoptics).Thisisnowourminimumspanningtreewithaminimumcostof: 50000+50000+30000+70000+30000+110000=$340000

Kruskal’s Algorithm cont...

When using Kruskal’s Algorithm, if there are two equal cheapest links to choose from, it will not matter which one you select. The same minimum spanning tree will be generated but in a different order.

Kruskal’s Algorithm is often called a ‘Greedy Algorithm’. It repeatedly adds the next cheapest link that doesn’t produce a cycle. It was named after Joseph Kruskal an American mathematician who lived from 1928 – 2010.

E

7A

B

5

3

5

G

C

3 D

11




WebeginbyfindingtheshortestpieceoftrackwhichisBtoD,withaweightof5(500metres).

D

13

F3

E

9

A

B C

G

3

5

5

cont...

Thecompletedminimumspanningtreehasatotalminimumweightof5+5+3+9+3+13=38.

ExampleAfarmerwishestodevelopasystemoftracksaroundhisfarmsothathe

canserviceallpartsofthefarm.Thedistanceinhundredsofmetresbetweenkeypointsonthefarmisgiveninthetablebelow.DrawupagraphfirstrepresentingthissituationandthenuseKruskal’sAlgorithmtofindaminimumspanningtreeforthefarmersohecandevelopasystemoftrackscoveringtheminimumdistance.

A B C D E F G HA - - - 9 8 - - -B - - 7 5 - - - -C - 7 - 8 - - 6 -D 9 5 8 - 10 12 11 13E 8 - - 10 - 15F - - 12 - - 8 10G - - 6 11 - 8 - -H - - - 13 15 10 - -

Drawingupagraphusingthedistancesfromthetablewegetthefollowing.

D

F

H

A

B

G

E

C

8

910 15

8

12116

8

57

10

13

D

B

5

ThenextshortestpieceoftrackisCtoGwithadistanceof600metressoweincludethislink.

D

B

G

C

6

5

ThenextshortestpieceoftrackisCtoBwithadistanceof700metressoweaddthislink.

D

B

G

C

6

57

ThenextshortestpiecesoftrackareCtoD,AtoEandGtoFallwithadistanceof800metres.WecannotaddCtoDaswewouldformacyclewithinthetreesowechoosetoaddGtoFandthenAtoE.

D

F

A

B

G

E

C

8

8

6

57

ThenextshortestpiecesoftrackisDtoAwithadistanceof900metressoweaddthislink.




64. Aplumberhastorunpipesfroma seriesofninepointsinabuilding. Atableoftheestimatedtime,inhours,tolay thepipesbetweeneachpointisgivenbelow. Firstdrawupagraphrepresentingthis situationandthenuseKruskal’sAlgorithmto findaminimumspanningtreeforthe pipingandthencalculateitsminimum layingtime.

A B C D E F G H IA - 5 - - - - - 9 -B 5 - 9 - - - - 12 -C - 9 - 8 - 5 - - 3D - - 8 - 10 15 - - -E - - - 10 - 11 - - -F - - 5 15 11 - 3 - -G - - - - - 3 - 2 7H 9 12 - - - - 2 - 8I - - 3 - - - 7 8 -

65. AinstallerhastolayTVcablingtoa neighbourhoodofninehouses. Atableoftheestimatedtimes,inhours, tolaythecablingbetweeneachhouseis givenbelow. Firstdrawupagraphrepresentingthis situationandthenuseKruskal’sAlgorithmto findaminimumspanningtreeforthe cablingandthencalculateitsminimum installingtime.

A B C D E F G H IA - 10 - 11 15 - - - -B 10 - 17 - 3 9 - - -C - 17 - - - 12 - - -D 11 - - - 14 - 21 - -E 15 3 - 14 - 16 19 4 -F - 9 12 - 16 - - 8 2G - - - 21 19 - - 5 -H - - - - 4 8 5 - 7I - - - - - 2 - 7 -©

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b) AlargewaterreservoiratvillageAisto supplywatertoallthevillages.Use Kruskal’salgorithmtofindaminimum spanningtreeforthenetworkofpipesand calculatethetotalamountofpiping required.

B F

A

C

E

G

I

D H

7.5km

10km

12.5km

4.2km 5km

4.2km3.3km 5km

7.5km 5.8km5km5.8km

2.5km 2.5km

4.2km 6.7km 5.8km 4.2km

c) AtrafficofficerleavesfromvillageA inthemorningandmustdriveallthe roadsbetweenthevillagesaspartofhis ‘run’.Givearouteforthetrafficofficerso thathevisitseachroadonlyonceandends upatvillageI.

d) Whatisthetotaldistancethetrafficofficer travelsinadayifhereturnstovillageA usingthequickestpossibleroute?




Page 1222. Evenvertices:A,B,C,D,E,F,G Oddvertices:None Classification:Eulercircuit Reason:Allverticesareeven.23. Evenvertices:A,C,E,F,G Oddvertices:B,D Classification:Eulerpath Reason:Hastwooddvertices andsomeevenvertices.24. Evenvertices:A,B Oddvertices:C,D Classification:Eulerpath Reason:Hastwooddvertices andsomeevenvertices.25. Evenvertices:C,D Oddvertices:A,B,E,F Classification:Neither Reason:Morethantwoodd verticessoneither.Page 1426. Evenvertices:A,B,C,D,E,F,G H,I Oddvertices:None Traversable:Yes StartandEndpoints:StartI EndI

A B

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Start/end

27. Evenvertices:C,D,E,H Oddvertices:A,B,F,G Traversable:No StartandEndpoints:N/A

D

C

E

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H

I

A B1

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11

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9 87

654

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2

Start/end

Page 1528. Evenvertices:A,B,C,D,E,F,G H,I Oddvertices:None Traversable:Yes StartandEndpoints:StartA EndA

29. Evenvertices:A,D,E,H Oddvertices:B,C,F,G,I,J Traversable:No StartandEndpoints:N/A

Page 1630. Evenvertices:A,B,C,D,E,G, Oddvertices:F,H Traversable:Yes StartandEndpoints:StartF EndH

31. Evenvertices:A,B,C,D,F,H,K Oddvertices:E,G,I,J Traversable:No StartandEndpoints:N/A

DC

EFG1

H

A B

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Start

End

Page 1732. Evenvertices:A,B,C,D,E,F,G H,I Oddvertices:None Traversable:Yes StartandEndpoints:StartB EndB

D CE

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A

BI

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87

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Start/end

33. Evenvertices:A,B,D,E,H,J Oddvertices:C,F,G,I Traversable:No StartandEndpoints:N/A

Page 2034. Circuit/path:Hamiltoniancircuit Justification:Possibletovisit eachvertexand returntothestart.

Path:C,E,F,D,A,B,C.35. Circuit/path:Hamiltoniancircuit Justification:Possibletovisit eachvertexand returntothestart.

Path:A,B,D,E,F,C,A.

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2Start/end

A B

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