C i at the rst not olored vertex, then in sequential order,

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K.Deshinkel AG44

Final Exam - 2h

1 Maximum height (/)

Thefollowingnetwork representsaportionofaityroad network.All theroadsaretwo-way

and representedby theedges between two destinations. Thenumberson the edgesindiate the

maximal heights(in entimeters)authorized for thevehiles.Adriverwants tomakeadelivery

frompointAtopointB.Insuhase,hewantstodeterminethemaximalheightx(inmeters)of

thetrukwhih isauthorized.

1. Showthatthisprobleminvolvestheresolutionofalassialproblemofgraph.Whih one?

2. Quoteandapplyanalgorithmtosolveaboveproblem.

3. Determinethemaximalheightofthetruk,suh thatthedeliveryispossibleandprovidea

feasibleroute.

2 Graph oloring(/)

A vertex-oloringproblem onsists in assigning alegalolor oneah vertexof agraph, suh

that, any two adjaent verties (onneted by an edge) are assigned two dierent olors. For a

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1. Drawtheundiretedgraphwitheightverties(A1 toA8),representedherebytheedges

(A1,A2), (A1,A3),(A1,A4),(A1,A7)

(A2,A3), (A2,A4),(A2,A5)

(A3,A5), (A3,A6),(A3,A7)

(A4,A5), (A4,A7)

(A5,A6), (A5,A7),(A5,A8)

(A6,A8)

(A7,A8)

2. The Welh-Powell algorithm gives an approximate solution in determining the hromati

number.Applythisalgorithmontheabovegraphandpresenttheresultofobtainednumber

ofolors.

Welh-Powellalgorithm

Step1:Orderthevertiesofthegraph

G

^by^the^dereasing^order^of^verties^degrees.

i = 1

Step 2 : Assign the olor

C ⁱ

ât ^the ^rst ^not ôlored ^vertex, ^then ⁱⁿ ^sequential ôrder,

assignolor

C ⁱ

^toêah^vertex^whihîs^notâdjaent^with^vertiesâlreadyôlored^with

C ⁱ

^.

i < −i + 1

Step3:Repeatthestep2untilallthevertiesareolored.

3. Explain whyabovegraphisnot2-olorable(ieusingonlytwoolors)?Determinethevalue

of

χ(G)

^for^this^graph.

4. Can you determine the hromati number of

χ(K ⁿ )

^, ^where

K ⁿ

^is ^a

n

^verties ^omplete

graph?

5. Givethetimeomplexity(usingO-notation)ofWelh-Powellalgorithmbasedon

n

^(number

ofverties)and

m

^(number^of^edges).

6. Theorem: thethree followingpropositionsareequivalent

G

^is2-olorable.

G

^is^a^bipartite^graph.

Everyylein

G

ôntainsânêven^numberôfêdges.

Givetheproofofthistheorem(equivaleneofpropositions)

7. ThefollowingTablesummarizestheandidates seletionexamination session,in whih the

linesorrespondtothevariousexaminationsubjetsandolumnstotheandidates.Ifeah

andidateisauthorizedtopartiipateonlyoneexamperday,howmanydaysarerequiredto

organizeanexaminationsessionthatallandidatesannishalltheexaminationsubjets?

Formulate the problem asavertex-oloringproblem. Speify the notation for theverties

andedgesin thegraph,themeaningofolors.Find asolutionforsuhproblem.

A B C D E F G H I J K L M

1 * * * * *

2 * * * *

3 * * * *

4 * *

5 * * * *

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roups dynamis is an interesting and important problem in soial psyhology. A group of

people and their relationships an be represented by a graph, in whih the verties represent

theindividuals andtheedges representtherelationsamong them.If thereis anedge onjoining

twoverties,thatindiatesthere isarelationshipbetweenthesetwoindividuals.Theproblemof

balaneinGroupdynamiswasintroduedbythepsyhologistHeider,whorepresentstheproblem

byasignedgraph.A signedgraphisagraph

G = (S, A)

^, ⁱⁿ^whih^a^positive⁽⁺⁾^or^negative^(-)

signisassignedoneahedgewhihindiatestheagreementordisagreementbetweentwoadjaent

individuals. A path or ayle in a signed graph is alled positive if it has an even number of

negativeedges,andtheoneontaininganoddnumbernegativeedgesisallednegative.Asigned

graphGissaidtobebalanedifandonlyifeveryyleofGispositive.

Example:

(a,d);(b,d);(b,)arethreepairsofindividualsinagreement

aandb areindisagreement,anddarein disagreement

therelationshipbetweenaandisnotknown

Thepathadbispositive,thepathadbdisnegativeTheyleadbaisnegative,theyleadba

ispositive.

1. Give(withoutonsideringisomorphigraphs)allompletesignedgraphswiththreeverties.

Indiateifitisbalanedornot.

2. Add sign

+/−

ôn^the êdges ôf âbove^graph^to ôbain ^three ^balaned ^graphsând ^three ^not

balaned graphs.

3. Indiate the balaned graphsamong the following graphs. Show for eah balaned graph

that itsvertexsetanbepartitionedintotwosubsets(oneofwhihmaybeempty)sothat

anyedgejoigningtwovertieswithinthesamesubsetispositivewhileanyedgejoigningtwo

vertiesin dierentsubsetsis negative.

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ofwhihmaybeempty)sothat anyedgejoigningtwovertieswithin thesamesubsetispositive

whileanyedgejoigningtwovertiesin dierentsubsetsisnegative.Provethetwotheorems.

(Theorem 1)A signedgraphGis balanedifand onlyifallpathsjoining thesamepairof

vertieshavethesamesign

(Theorem2) AsignedgraphGisbalanedifandonlyifeveryyleofGispositive

4 Inompatibility (/)

Agroupofeightpeopleismeetingfordinner.Theinompatiblerelationsamongthemembers

aresummarizedin thefollowingtable :

A B C D E F G H

doesnotagreewith B,D A,F,E,H D,E A,C,G B,C,F B,E,H H,D B,F,G

1. Model the onits among people in theform of agraph(speify the notationof verties

andedges).

2. Proposeaplantable(the tableisirular)forthisgroupbyseparatingtheanytwoinom-

patible persons.Show that theproblem onsists in ndinga partiularset of vertiesin a

partiulargraph(speifywhihsetin whihgraph).

5 Flow

Consider the direted network below. The numbers on the edges represent the apaity of

eah edge. Quote and apply an algorithm to nd amaximum ow in this network,from s to t

(Showatleastonestepofthealgorithm).Also,ndaminimumutinthenetwork.

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